5460:
1404:
2161:
1858:
590:
1170:
1676:
386:
1871:) fewer variables. This means that, in principle, one can restrict attention to convex optimization problems without equality constraints. In practice, however, it is often preferred to retain the equality constraints, since they might make some algorithms more efficient, and also make the problem easier to understand and analyze.
1540:
1883:
1399:{\displaystyle {\begin{aligned}&{\underset {\mathbf {x} ,t}{\operatorname {minimize} }}&&t\\&\operatorname {subject\ to} &&f(\mathbf {x} )-t\leq 0\\&&&g_{i}(\mathbf {x} )\leq 0,\quad i=1,\dots ,m\\&&&h_{i}(\mathbf {x} )=0,\quad i=1,\dots ,p,\end{aligned}}}
3449:
Solvers implement the algorithms themselves and are usually written in C. They require users to specify optimization problems in very specific formats which may not be natural from a modeling perspective. Modeling tools are separate pieces of software that let the user specify an optimization in
2632:
1853:{\displaystyle {\begin{aligned}&{\underset {\mathbf {x} }{\operatorname {minimize} }}&&f(\mathbf {F\mathbf {z} +\mathbf {x} _{0}} )\\&\operatorname {subject\ to} &&g_{i}(\mathbf {F\mathbf {z} +\mathbf {x} _{0}} )\leq 0,\quad i=1,\dots ,m\\\end{aligned}}}
585:{\displaystyle {\begin{aligned}&{\underset {\mathbf {x} }{\operatorname {minimize} }}&&f(\mathbf {x} )\\&\operatorname {subject\ to} &&g_{i}(\mathbf {x} )\leq 0,\quad i=1,\dots ,m\\&&&h_{i}(\mathbf {x} )=0,\quad i=1,\dots ,p,\end{aligned}}}
1424:
2454:
953:
689:
122:
2976:
3103:
2878:
2777:
2465:
263:
837:
746:
1681:
1429:
1175:
391:
3331:
633:
3203:
1065:
167:
206:
5960:
1535:{\displaystyle {\begin{aligned}&{\underset {\mathbf {x} }{\operatorname {minimize} }}&&c^{T}x\\&\operatorname {subject\ to} &&x\in (b+L)\cap K\end{aligned}}}
2089:
methods, which either find a feasible point or show that none exist. Phase I methods generally consist of reducing the search in question to a simpler convex optimization problem.
3012:
3576:
Interfaces with CPLEX, GUROBI, MOSEK, SDPT3, SEDUMI, CSDP, SDPA, PENNON solvers; also supports integer and nonlinear optimization, and some nonconvex optimization. Can perform
4859:, Dirk TM Slock, and Lisa Meilhac. "Online angle of arrival estimation in the presence of mutual coupling." 2016 IEEE Statistical Signal Processing Workshop (SSP). IEEE, 2016.
982:
3364:
3239:
2346:
2290:
1089:
5189:
2322:
875:
784:
2907:
4072:
3700:
1009:
2248:
1142:
3424:
3404:
3384:
3282:
3262:
3127:
2804:
2715:
2695:
2675:
2655:
2354:
1119:
1033:
361:
334:
314:
5953:
5357:
1160:. This is because any program with a general objective can be transformed into a program with a linear objective by adding a single variable t and a single
3453:
The table below shows a mix of modeling tools (such as CVXPY and Convex.jl) and solvers (such as CVXOPT and MOSEK). This table is by no means exhaustive.
2054:
can be used. It can be seen as reducing a general unconstrained convex problem, to a sequence of quadratic problems.Newton's method can be combined with
2077:
The more challenging problems are those with inequality constraints. A common way to solve them is to reduce them to unconstrained problems by adding a
5132:
4431:
5946:
5228:
4254:
3662:
38:
over convex sets). Many classes of convex optimization problems admit polynomial-time algorithms, whereas mathematical optimization is in general
4516:
Peng, Jiming; Roos, Cornelis; Terlaky, Tamás (2002). "Self-regular functions and new search directions for linear and semidefinite optimization".
4362:
1879:
The following problem classes are all convex optimization problems, or can be reduced to convex optimization problems via simple transformations:
884:
1951:
641:
74:
5352:
4823:
1570:
It is possible to convert a convex program in standard form, to a convex program with no equality constraints. Denote the equality constraints
4076:
4856:
4785:
4701:
Udell, Madeleine; Mohan, Karanveer; Zeng, David; Hong, Jenny; Diamond, Steven; Boyd, Stephen (2014-10-17). "Convex
Optimization in Julia".
2913:
3989:, where the approximation concept has proven to be efficient. Convex optimization can be used to model problems in the following fields:
3018:
2809:
2627:{\displaystyle L(x,\lambda _{0},\lambda _{1},\ldots ,\lambda _{m})=\lambda _{0}f(x)+\lambda _{1}g_{1}(x)+\cdots +\lambda _{m}g_{m}(x).}
6150:
2720:
3450:
higher-level syntax. They manage all transformations to and from the user's high-level model and the solver's input/output format.
2139:
Subgradient methods can be implemented simply and so are widely used. Dual subgradient methods are subgradient methods applied to a
4227:
2102:
2096:
2032:
problems, or the problems with only equality constraints. As the equality constraints are all linear, they can be eliminated with
6050:
5846:
5366:
5196:
4645:
214:
4847:
Ben Haim Y. and
Elishakoff I., Convex Models of Uncertainty in Applied Mechanics, Elsevier Science Publishers, Amsterdam, 1990
6142:
5221:
5096:
5066:
5035:
5004:
4925:
4903:
4881:
4579:
4321:
4093:
792:
701:
2182:
5081:
Computational combinatorial optimization: Papers from the Spring School held in Schloß Dagstuhl, May 15–19, 2000
1098:
Many optimization problems can be equivalently formulated in this standard form. For example, the problem of maximizing a
5927:
5389:
4108:
3856:
Supports general-purpose codes for LP + SOCP + SDP. Uses a bundle method. Special support for SDP and SOCP constraints.
3825:
Solves LP + SDP. Supports primal-dual methods for LP + SDP. Parallelized and extended precision versions are available.
2039:
In the class of unconstrained (or equality-constrained) problems, the simplest ones are those in which the objective is
6399:
6155:
5441:
5302:
4500:
3290:
2051:
601:
5409:
4968:
4264:
4237:
3135:
2208:
4473:
For methods for convex minimization, see the volumes by
Hiriart-Urruty and Lemaréchal (bundle) and the textbooks by
2190:
1038:
5520:
5214:
3931:
Modeling system for linear, nonlinear, mixed integer linear/nonlinear, and second-order cone programming problems.
3901:
Supports general-purpose codes. Uses an augmented
Lagrangian method, especially for problems with SDP constraints.
3733:
Supports primal-dual methods for LP + SOCP. Can solve LP, QP, SOCP, and mixed integer linear programming problems.
3643:
2050:
For unconstrained (or equality-constrained) problems with a general convex objective that is twice-differentiable,
1920:
are the next-simplest. In QP, the constraints are all linear, but the objective may be a convex quadratic function.
1144:. The problem of maximizing a concave function over a convex set is commonly called a convex optimization problem.
5459:
6175:
6092:
137:
4936:
5797:
2186:
2036:
and integrated into the objective, thus converting an equality-constrained problem into an unconstrained one.
176:
5030:. Grundlehren der Mathematischen Wissenschaften . Vol. 306. Berlin: Springer-Verlag. pp. xviii+346.
4999:. Grundlehren der Mathematischen Wissenschaften . Vol. 305. Berlin: Springer-Verlag. pp. xviii+417.
5905:
5525:
3502:
3470:
2008:
1923:
1895:
3773:
Supports primal-dual methods for LP + SOCP + SDP. Uses
Nesterov-Todd scaling. Interfaces to MOSEK and DSDP.
6394:
5841:
5809:
4010:
3538:
6195:
5154:
4474:
4141:
Murty, Katta; Kabadi, Santosh (1987). "Some NP-complete problems in quadratic and nonlinear programming".
5890:
5515:
3642:
Can do robust optimization on linear programming (with MOSEK to solve second-order cone programming) and
2981:
1914:
problems are the simplest convex programs. In LP, the objective and constraint functions are all linear.
1418:, which means minimizing a linear objective over the intersection of an affine plane and a convex cone:
6160:
5836:
5792:
5685:
5414:
5394:
4313:
4033:
2066:
2004:
1152:
In the standard form it is possible to assume, without loss of generality, that the objective function
1095:
of convex functions are convex, affine sets are convex, and the intersection of convex sets is convex.
5604:
958:
6200:
6190:
5575:
5237:
4021:
3434:
There is a large software ecosystem for convex optimization. This ecosystem has two main categories:
23:
5760:
5206:
4816:
4451:
3961:
Convex optimization can be used to model problems in a wide range of disciplines, such as automatic
3336:
3211:
2327:
2253:
2147:
method is similar to the dual subgradient method, but takes a time average of the primal variables.
1070:
6389:
6180:
6165:
6007:
5622:
4103:
3752:
3746:
3556:
2295:
2171:
2112:
1929:
1899:
1161:
842:
751:
1605:
is infeasible, then of course the original problem is infeasible. Otherwise, it has some solution
1559:. A linear program in standard form in the special case in which K is the nonnegative orthant of R
6250:
6227:
6121:
6045:
5804:
5703:
5419:
4778:
3986:
3916:
Supports general-purpose codes. Uses low-rank factorization with an augmented
Lagrangian method.
3489:
Interfaces with SeDuMi and SDPT3 solvers; designed to only express convex optimization problems.
2175:
5083:. Lecture Notes in Computer Science. Vol. 2241. Berlin: Springer-Verlag. pp. 112–156.
4379:
6368:
6329:
6245:
6170:
6097:
6082:
6035:
5895:
5880:
5770:
5648:
5297:
5274:
5241:
5140:
5028:
Convex analysis and minimization algorithms, Volume II: Advanced theory and bundle methods
4446:
4185:
Sahni, S. "Computationally related problems," in SIAM Journal on
Computing, 3, 262--279, 1974.
4088:
4060:
4027:
3993:
2140:
2108:
2082:
6314:
6306:
6302:
6298:
6294:
6290:
1999:
These results are used by the theory of convex minimization along with geometric notions from
6102:
5784:
5750:
5653:
5595:
5476:
5282:
5262:
4056:
3886:
Supports general-purpose codes for SOCP, which it treats as a nonlinear programming problem.
3596:
Expresses and solves semidefinite programming problems (called "linear matrix inequalities")
2883:
2118:
2040:
1956:
1917:
1891:
878:
6107:
5973:
5938:
5831:
5658:
5570:
5106:
5045:
5014:
4913:
4891:
4559:
4098:
3661:
Modeling system for robust optimization. Supports distributionally robust optimization and
3285:
2780:
2081:, enforcing the inequality constraints, to the objective function. Such methods are called
1961:
987:
5076:
5023:
4992:
4980:
4256:
Lectures on modern convex optimization: analysis, algorithms, and engineering applications
2449:{\displaystyle {\mathcal {X}}=\left\{x\in X\vert g_{1}(x),\ldots ,g_{m}(x)\leq 0\right\}.}
2224:
8:
6040:
6025:
5900:
5765:
5718:
5708:
5560:
5361:
5344:
5249:
4014:
4006:
3577:
2047:(which are necessary for optimality) are all linear, so they can be solved analytically.
2000:
5201:
4304:
1124:
6269:
5987:
5635:
5590:
5580:
5371:
5287:
5110:
5055:
4702:
4541:
4412:
4394:
4196:
4168:
3409:
3389:
3369:
3267:
3247:
3112:
2789:
2700:
2680:
2660:
2640:
2144:
2133:
2128:
2092:
Convex optimization problems can also be solved by the following contemporary methods:
2058:
for an appropriate step size, and it can be mathematically proven to converge quickly.
1935:
1911:
1903:
1887:
1104:
1018:
346:
319:
299:
6087:
5643:
5321:
5092:
5062:
5031:
5000:
4964:
4921:
4899:
4877:
4652:
4575:
4533:
4496:
4478:
4317:
4260:
4233:
3966:
2085:.They have to be initialized by finding a feasible interior point using by so-called
35:
4545:
4416:
4172:
6240:
6185:
6076:
6071:
5723:
5713:
5617:
5494:
5399:
5381:
5334:
5245:
5114:
5084:
4567:
4525:
4456:
4404:
4208:
4158:
4150:
2123:
2078:
2062:
2012:
1121:
can be re-formulated equivalently as the problem of minimizing the convex function
1099:
1067:
satisfying the inequality and the equality constraints. This set is convex because
4408:
6260:
6231:
6205:
6126:
6111:
6020:
5992:
5969:
5739:
5102:
5041:
5010:
4958:
4068:
3962:
2221:
Consider a convex minimization problem given in standard form by a cost function
1550:
1157:
692:
64:
27:
3871:
Supports general-purpose codes for LP + SDP. Uses a dual interior point method.
6347:
6255:
6116:
6015:
5727:
5612:
5499:
5433:
5404:
4938:
Convex
Analysis and Nonlinear Optimization: Theory and Examples, Second Edition
3982:
3970:
2044:
2033:
1982:
4571:
3718:
Modeling system for polynomial optimization. Uses the SDPA or SeDuMi solvers.
336:, or the infimum is not attained, then the optimization problem is said to be
6383:
6131:
5885:
5869:
4537:
4344:
1978:
1946:
132:
5088:
4378:
Agrawal, Akshay; Verschueren, Robin; Diamond, Steven; Boyd, Stephen (2018).
3751:
Supports primal-dual methods for LP + SDP. Interfaces available for MATLAB,
948:{\displaystyle h_{i}(\mathbf {x} )=\mathbf {a_{i}} \cdot \mathbf {x} -b_{i}}
6352:
5823:
5329:
5079:(2001). "Lagrangian relaxation". In Michael JĂĽnger and Denis Naddef (ed.).
1882:
1092:
269:
In general, there are three options regarding the existence of a solution:
4529:
3841:
Solves LP + SOCP + SDP. Supports primal-dual methods for LP + SOCP + SDP.
3807:
Solves LP + SOCP + SDP. Supports primal-dual methods for LP + SOCP + SDP.
684:{\displaystyle f:{\mathcal {D}}\subseteq \mathbb {R} ^{n}\to \mathbb {R} }
117:{\displaystyle f:{\mathcal {D}}\subseteq \mathbb {R} ^{n}\to \mathbb {R} }
6342:
6337:
6221:
5910:
5292:
5183:
5179:
5167:
Structural synthesis by combining approximation concepts and dual methods
4756:
Structural synthesis by combining approximation concepts and dual methods
4064:
4040:
2055:
1546:
4997:
Convex analysis and minimization algorithms, Volume I: Fundamentals
4163:
6265:
6235:
5997:
4212:
4154:
3978:
31:
1147:
5236:
4956:
2971:{\displaystyle \lambda _{0},\lambda _{1},\ldots ,\lambda _{m}\geq 0,}
1973:
The following are useful properties of convex optimization problems:
4460:
2160:
6274:
5312:
5202:
Convex
Optimization Book by Lieven Vandenberghe and Stephen P. Boyd
4399:
3802:
3098:{\displaystyle \lambda _{1}g_{1}(x)=\cdots =\lambda _{m}g_{m}(x)=0}
2873:{\displaystyle L(y,\lambda _{0},\lambda _{1},\ldots ,\lambda _{m})}
4707:
2023:
6055:
5632:
3974:
39:
4872:
Bertsekas, Dimitri P.; Nedic, Angelia; Ozdaglar, Asuman (2003).
4777:
Boyd, Stephen; Diamond, Stephen; Zhang, Junzi; Agrawal, Akshay.
4564:
Springer Series in
Operations Research and Financial Engineering
3703:. Uses SDPT3 and SeDuMi. Requires Symbolic Computation Toolbox.
4722:
4197:"Quadratic programming with one negative eigenvalue is NP-hard"
3946:
XML standard for encoding optimization problems and solutions.
3484:
2772:{\displaystyle \lambda _{0},\lambda _{1},\ldots ,\lambda _{m},}
4817:"Convex optimization: applications, formulations, relaxations"
4055:
Extensions of convex optimization include the optimization of
3109:
If there exists a "strictly feasible point", that is, a point
2061:
Other efficient algorithms for unconstrained minimization are
55:
A convex optimization problem is defined by two ingredients:
4377:
3820:
3208:
then the statement above can be strengthened to require that
5968:
4364:
Interior point polynomial-time methods in convex programming
5061:. Lecture Notes in Mathematics. New York: Springer-Verlag.
4493:
Interior-Point Polynomial Algorithms in Convex Programming
4226:
Hiriart-Urruty, Jean-Baptiste; Lemaréchal, Claude (1996).
1565:
4677:
4253:
Ben-Tal, Aharon; NemirovskiÄ, ArkadiÄ Semenovich (2001).
4229:
Convex analysis and minimization algorithms: Fundamentals
258:{\displaystyle \inf\{f(\mathbf {x} ):\mathbf {x} \in C\}}
5021:
4990:
4963:. Vol. 153. Springer Science & Business Media.
4225:
1952:
Quadratic minimization with convex quadratic constraints
5124:
Interior Point Polynomial Methods in Convex Programming
4429:
3543:
Disciplined convex programming, supports many solvers.
1995:
convex, then the problem has at most one optimal point.
4871:
3765:
3755:, and Python. Parallel version available. SDP solver.
3508:
2115:
barrier functions and self-regular barrier functions.
832:{\displaystyle h_{i}:\mathbb {R} ^{n}\to \mathbb {R} }
741:{\displaystyle g_{i}:\mathbb {R} ^{n}\to \mathbb {R} }
5057:
Methods of Descent for Nondifferentiable Optimization
4380:"A rewriting system for convex optimization problems"
3646:. Modeling package for LP + SDP and robust versions.
3412:
3392:
3372:
3339:
3293:
3270:
3250:
3214:
3138:
3115:
3021:
2984:
2916:
2886:
2812:
2792:
2723:
2703:
2683:
2663:
2643:
2468:
2357:
2330:
2298:
2256:
2227:
1679:
1427:
1173:
1127:
1107:
1073:
1041:
1021:
990:
961:
887:
845:
795:
754:
704:
644:
604:
389:
349:
322:
302:
217:
179:
140:
77:
5749:
1612:, and the set of all solutions can be presented as:
5121:
4776:
4678:"Welcome to CVXPY 1.1 — CVXPY 1.1.11 documentation"
4129:
2134:
Dual subgradients and the drift-plus-penalty method
1148:
Epigraph form (standard form with linear objective)
1035:of the optimization problem consists of all points
5169:. J. Amer. Inst. Aeronaut. Astronaut 18, 1252-1260
5054:
4758:. J. Amer. Inst. Aeronaut. Astronaut 18, 1252-1260
4700:
4495:. Society for Industrial and Applied Mathematics.
4345:"Optimization Problem Types - Convex Optimization"
4252:
3418:
3398:
3378:
3358:
3325:
3276:
3256:
3233:
3197:
3121:
3097:
3006:
2970:
2901:
2872:
2798:
2771:
2709:
2689:
2669:
2649:
2626:
2448:
2340:
2316:
2284:
2242:
1886:A hierarchy of convex optimization problems. (LP:
1852:
1534:
1398:
1136:
1113:
1083:
1059:
1027:
1003:
976:
947:
869:
831:
778:
740:
683:
627:
584:
355:
328:
308:
257:
200:
161:
116:
4646:"An Overview Of Software For Convex Optimization"
4360:
3326:{\displaystyle \lambda _{0},\ldots ,\lambda _{m}}
363:is the empty set, then the problem is said to be
6381:
4957:Christensen, Peter W.; Anders Klarbring (2008).
4302:
4195:Pardalos, Panos M.; Vavasis, Stephen A. (1991).
4071:and iterative methods for approximately solving
3627:Similar to LMI lab, but uses the SeDuMi solver.
2783:, that satisfy these conditions simultaneously:
1938:are even more general - see figure to the right,
628:{\displaystyle \mathbf {x} \in \mathbb {R} ^{n}}
218:
5197:An overview of software for convex optimization
4515:
4490:
4194:
4125:
4123:
3611:Transforms LMI lab problems into SDP problems.
3198:{\displaystyle g_{1}(z),\ldots ,g_{m}(z)<0,}
2024:Unconstrained and equality-constrained problems
5122:Nesterov, Yurii; Nemirovskii, Arkadii (1994).
4491:Nesterov, Yurii; Arkadii, Nemirovskii (1995).
4481:, and Boyd and Vandenberghe (interior point).
1060:{\displaystyle \mathbf {x} \in {\mathcal {D}}}
285:; the set of all optimal points is called the
5954:
5222:
4423:
3684:Modeling system for polynomial optimization.
3580:with uncertainty in LP/SOCP/SDP constraints.
2028:The convex programs easiest to solve are the
5783:
5133:Introductory Lectures on Convex Optimization
4934:
4509:
4303:Boyd, Stephen; Vandenberghe, Lieven (2004).
4120:
3788:Supports primal-dual methods for LP + SOCP.
2382:
252:
221:
5261:
5153:
5139:
4140:
2459:The Lagrangian function for the problem is
2189:. Unsourced material may be challenged and
1414:Every convex program can be presented in a
162:{\displaystyle C\subseteq \mathbb {R} ^{n}}
5961:
5947:
5229:
5215:
5075:
4960:An introduction to structural optimization
4079:, also known as abstract convex analysis.
3969:, communications and networks, electronic
5475:
4935:Borwein, Jonathan; Lewis, Adrian (2000).
4912:
4890:
4706:
4450:
4398:
4371:
4162:
2209:Learn how and when to remove this message
825:
811:
734:
720:
677:
663:
615:
201:{\displaystyle \mathbf {x^{\ast }} \in C}
149:
110:
96:
5463:Optimization computes maxima and minima.
5147:. Princeton: Princeton University Press.
1881:
635:is the vector of optimization variables;
173:The goal of the problem is to find some
6051:Locally convex topological vector space
5547:
5190:6.253: Convex Analysis and Optimization
4723:"Disciplined Convex Optimiation - CVXR"
4067:functions. Extensions of the theory of
2150:
1566:Eliminating linear equality constraints
26:that studies the problem of minimizing
6382:
5052:
4814:
4639:
4637:
4635:
4633:
4631:
4629:
4627:
4625:
4623:
4621:
4619:
4617:
4615:
4613:
4611:
5942:
5867:
5683:
5659:Principal pivoting algorithm of Lemke
5546:
5474:
5260:
5210:
4810:
4808:
4806:
4772:
4770:
4768:
4766:
4764:
4609:
4607:
4605:
4603:
4601:
4599:
4597:
4595:
4593:
4591:
4432:"Lagrange multipliers and optimality"
4356:
4354:
4298:
4296:
16:Subfield of mathematical optimization
4643:
4294:
4292:
4290:
4288:
4286:
4284:
4282:
4280:
4278:
4276:
4046:Localization using wireless signals
2187:adding citations to reliable sources
2154:
698:The inequality constraint functions
376:A convex optimization problem is in
4979:Hiriart-Urruty, Jean-Baptiste, and
4920:. Belmont, MA.: Athena Scientific.
4898:. Belmont, MA.: Athena Scientific.
4876:. Belmont, MA.: Athena Scientific.
4109:Algorithmic problems on convex sets
2072:
13:
5868:
5458:
5303:Successive parabolic interpolation
4803:
4779:"Convex Optimization Applications"
4761:
4588:
4351:
3007:{\displaystyle \lambda _{k}>0,}
2360:
2333:
1767:
1764:
1758:
1755:
1752:
1749:
1746:
1743:
1740:
1494:
1491:
1485:
1482:
1479:
1476:
1473:
1470:
1467:
1238:
1235:
1229:
1226:
1223:
1220:
1217:
1214:
1211:
1076:
1052:
789:The equality constraint functions
653:
457:
454:
448:
445:
442:
439:
436:
433:
430:
277:* exists, it is referred to as an
86:
14:
6411:
5684:
5623:Projective algorithm of Karmarkar
5173:
4273:
5618:Ellipsoid algorithm of Khachiyan
5521:Sequential quadratic programming
5358:Broyden–Fletcher–Goldfarb–Shanno
4874:Convex Analysis and Optimization
4430:Rockafellar, R. Tyrrell (1993).
3644:mixed integer linear programming
2159:
2099:(Wolfe, Lemaréchal, Kiwiel), and
2003:(in Hilbert spaces) such as the
1874:
1806:
1801:
1796:
1792:
1788:
1725:
1720:
1715:
1711:
1707:
1691:
1439:
1351:
1292:
1252:
1186:
1043:
977:{\displaystyle \mathbf {a_{i}} }
968:
964:
928:
918:
914:
902:
606:
537:
478:
417:
401:
371:
242:
231:
182:
50:
6156:Ekeland's variational principle
5165:Schmit, L.A.; Fleury, C. 1980:
5022:Hiriart-Urruty, Jean-Baptiste;
4991:Hiriart-Urruty, Jean-Baptiste;
4985:Fundamentals of Convex analysis
4850:
4841:
4829:from the original on 2021-04-12
4791:from the original on 2015-10-01
4754:Schmit, L.A.; Fleury, C. 1980:
4748:
4739:
4715:
4694:
4670:
4552:
4484:
4467:
4130:Nesterov & Nemirovskii 1994
4075:problems occur in the field of
4039:Non-probabilistic modelling of
3956:
1824:
1670:in the original problem gives:
1367:
1308:
553:
494:
5576:Reduced gradient (Frank–Wolfe)
5184:EE364b: Convex Optimization II
4918:Convex Optimization Algorithms
4745:Chritensen/Klarbring, chpt. 4.
4337:
4246:
4219:
4201:Journal of Global Optimization
4188:
4179:
4134:
3973:, data analysis and modeling,
3359:{\displaystyle \lambda _{0}=1}
3234:{\displaystyle \lambda _{0}=1}
3183:
3177:
3155:
3149:
3086:
3080:
3048:
3042:
2867:
2816:
2618:
2612:
2580:
2574:
2548:
2542:
2523:
2472:
2429:
2423:
2401:
2395:
2341:{\displaystyle {\mathcal {X}}}
2285:{\displaystyle g_{i}(x)\leq 0}
2273:
2267:
2237:
2231:
1812:
1784:
1731:
1703:
1519:
1507:
1355:
1347:
1296:
1288:
1256:
1248:
1084:{\displaystyle {\mathcal {D}}}
906:
898:
821:
730:
673:
541:
533:
482:
474:
421:
413:
235:
227:
106:
34:(or, equivalently, maximizing
1:
5906:Spiral optimization algorithm
5526:Successive linear programming
5180:EE364a: Convex Optimization I
5161:. Princeton University Press.
5136:, Kluwer Academic Publishers
5053:Kiwiel, Krzysztof C. (1985).
4865:
4815:Malick, JĂ©rĂ´me (2011-09-28).
4409:10.1080/23307706.2017.1397554
4094:Karush–Kuhn–Tucker conditions
4050:
2317:{\displaystyle 1\leq i\leq m}
2018:
2009:separating hyperplane theorem
1991:if the objective function is
1968:
1964:with appropriate constraints.
1942:Other special cases include;
1924:Second order cone programming
1409:
870:{\displaystyle i=1,\ldots ,p}
779:{\displaystyle i=1,\ldots ,m}
45:
5644:Simplex algorithm of Dantzig
5516:Augmented Lagrangian methods
5192:, an MIT OCW course homepage
4020:Model fitting (particularly
289:; and the problem is called
7:
6176:Hermite–Hadamard inequality
5186:, Stanford course homepages
4082:
3983:optimal experimental design
3429:
2717:, there exist real numbers
2250:and inequality constraints
1867:. Note that there are rank(
10:
6416:
4896:Convex Optimization Theory
4361:Arkadi Nemirovsky (2004).
4314:Cambridge University Press
4034:Combinatorial optimization
3105:(complementary slackness).
2043:. For these problems, the
2005:Hilbert projection theorem
1988:the optimal set is convex;
6400:Mathematical optimization
6361:
6328:
6283:
6214:
6140:
6064:
6006:
5980:
5923:
5876:
5863:
5847:Push–relabel maximum flow
5822:
5738:
5696:
5692:
5679:
5649:Revised simplex algorithm
5631:
5603:
5589:
5559:
5555:
5542:
5508:
5487:
5483:
5470:
5456:
5432:
5380:
5343:
5320:
5311:
5273:
5269:
5256:
5130:Nesterov, Yurii. (2004).
4572:10.1007/978-0-387-40065-5
4232:. Springer. p. 291.
4022:multiclass classification
3999:Worst-case risk analysis.
1896:second-order cone program
63:, which is a real-valued
24:mathematical optimization
6362:Applications and related
6166:Fenchel-Young inequality
5372:Symmetric rank-one (SR1)
5353:Berndt–Hall–Hall–Hausman
4560:"Numerical Optimization"
4518:Mathematical Programming
4143:Mathematical Programming
4114:
4104:Proximal gradient method
1930:Semidefinite programming
1900:semidefinite programming
1863:where the variables are
1556:, and b is a vector in R
881:, that is, of the form:
316:is unbounded below over
6122:Legendre transformation
6046:Legendre transformation
5896:Parallel metaheuristics
5704:Approximation algorithm
5415:Powell's dog leg method
5367:Davidon–Fletcher–Powell
5263:Unconstrained nonlinear
5089:10.1007/3-540-45586-8_4
4073:non-convex minimization
3987:structural optimization
3701:polynomial optimization
3386:is certain to minimize
2902:{\displaystyle y\in X,}
786:, are convex functions;
638:The objective function
6369:Convexity in economics
6303:(lower) ideally convex
6161:Fenchel–Moreau theorem
6151:Carathéodory's theorem
5881:Evolutionary algorithm
5464:
5159:Nonlinear Optimization
4028:Electricity generation
4007:statistical regression
3994:Portfolio optimization
3941:Optimization Services
3420:
3400:
3380:
3360:
3327:
3284:satisfies (1)–(3) for
3278:
3258:
3235:
3199:
3123:
3099:
3008:
2972:
2903:
2874:
2800:
2773:
2711:
2691:
2671:
2651:
2628:
2450:
2342:
2318:
2286:
2244:
2109:Interior-point methods
2103:Subgradient projection
2083:interior point methods
1907:
1861:
1854:
1536:
1400:
1138:
1115:
1085:
1061:
1029:
1005:
978:
949:
879:affine transformations
871:
833:
780:
742:
685:
629:
586:
357:
330:
310:
259:
202:
163:
118:
6291:Convex series related
6191:Shapley–Folkman lemma
5654:Criss-cross algorithm
5477:Constrained nonlinear
5462:
5283:Golden-section search
4914:Bertsekas, Dimitri P.
4892:Bertsekas, Dimitri P.
4530:10.1007/s101070200296
4077:generalized convexity
3446:) on the other hand.
3421:
3401:
3381:
3361:
3328:
3279:
3259:
3236:
3200:
3124:
3100:
3009:
2973:
2904:
2875:
2801:
2774:
2712:
2692:
2672:
2652:
2629:
2451:
2343:
2319:
2287:
2245:
2119:Cutting-plane methods
1957:Geometric programming
1918:Quadratic programming
1892:quadratic programming
1885:
1855:
1672:
1655:matrix. Substituting
1537:
1401:
1139:
1116:
1086:
1062:
1030:
1006:
1004:{\displaystyle b_{i}}
979:
950:
872:
834:
781:
743:
686:
630:
587:
358:
331:
311:
260:
203:
164:
119:
6181:Krein–Milman theorem
5974:variational analysis
5571:Cutting-plane method
5155:Ruszczyński, Andrzej
4387:Control and Decision
4259:. pp. 335–336.
4099:Optimization problem
4002:Optimal advertising.
3838:MATLAB, Octave, MEX
3699:Modeling system for
3507:Interfaces with the
3438:on the one hand and
3410:
3390:
3370:
3337:
3291:
3268:
3248:
3244:Conversely, if some
3212:
3136:
3113:
3019:
2982:
2914:
2884:
2810:
2790:
2781:Lagrange multipliers
2721:
2701:
2681:
2661:
2641:
2466:
2355:
2328:
2296:
2254:
2243:{\displaystyle f(x)}
2225:
2183:improve this section
2151:Lagrange multipliers
2111:, which make use of
1962:Entropy maximization
1677:
1545:where K is a closed
1425:
1171:
1125:
1105:
1071:
1039:
1019:
988:
959:
885:
843:
793:
752:
702:
642:
602:
387:
380:if it is written as
347:
320:
300:
215:
177:
138:
75:
6395:Convex optimization
6171:Jensen's inequality
6041:Lagrange multiplier
6031:Convex optimization
6026:Convex metric space
5901:Simulated annealing
5719:Integer programming
5709:Dynamic programming
5549:Convex optimization
5410:Levenberg–Marquardt
4987:. Berlin: Springer.
4306:Convex Optimization
4015:quantile regression
3578:robust optimization
3457:
2065:(a special case of
2001:functional analysis
1547:pointed convex cone
20:Convex optimization
6299:(cs, bcs)-complete
6270:Algebraic interior
5988:Convex combination
5581:Subgradient method
5465:
5390:Conjugate gradient
5298:Nelder–Mead method
5141:Rockafellar, R. T.
5077:Lemaréchal, Claude
5024:Lemaréchal, Claude
4993:Lemaréchal, Claude
4981:Lemaréchal, Claude
4213:10.1007/BF00120662
4155:10.1007/BF02592948
3606:LMIlab translator
3456:
3416:
3396:
3376:
3356:
3323:
3274:
3254:
3231:
3195:
3119:
3095:
3004:
2978:with at least one
2968:
2899:
2870:
2796:
2769:
2707:
2687:
2667:
2647:
2624:
2446:
2338:
2324:. Then the domain
2314:
2282:
2240:
2145:drift-plus-penalty
2129:Subgradient method
1936:Conic optimization
1912:Linear programming
1908:
1904:conic optimization
1888:linear programming
1850:
1848:
1695:
1532:
1530:
1443:
1396:
1394:
1197:
1137:{\displaystyle -f}
1134:
1111:
1081:
1057:
1025:
1001:
974:
945:
867:
829:
776:
738:
681:
625:
582:
580:
405:
353:
326:
306:
255:
198:
159:
114:
61:objective function
6377:
6376:
5936:
5935:
5919:
5918:
5859:
5858:
5855:
5854:
5818:
5817:
5779:
5778:
5675:
5674:
5671:
5670:
5667:
5666:
5538:
5537:
5534:
5533:
5454:
5453:
5450:
5449:
5428:
5427:
5098:978-3-540-42877-0
5068:978-3-540-15642-0
5037:978-3-540-56852-0
5006:978-3-540-56850-6
4927:978-1-886529-28-1
4905:978-1-886529-31-1
4883:978-1-886529-45-8
4644:Borchers, Brian.
4581:978-0-387-30303-1
4347:. 9 January 2011.
4323:978-0-521-83378-3
3967:signal processing
3965:, estimation and
3954:
3953:
3419:{\displaystyle X}
3399:{\displaystyle f}
3379:{\displaystyle x}
3277:{\displaystyle X}
3257:{\displaystyle x}
3122:{\displaystyle z}
2799:{\displaystyle x}
2710:{\displaystyle X}
2690:{\displaystyle f}
2670:{\displaystyle X}
2650:{\displaystyle x}
2219:
2218:
2211:
2105:methods (Polyak),
1932:are more general.
1926:are more general.
1763:
1686:
1490:
1434:
1234:
1180:
1114:{\displaystyle f}
1028:{\displaystyle C}
1015:The feasible set
453:
396:
356:{\displaystyle C}
329:{\displaystyle C}
309:{\displaystyle f}
36:concave functions
22:is a subfield of
6407:
6295:(cs, lcs)-closed
6241:Effective domain
6196:Robinson–Ursescu
6072:Convex conjugate
5963:
5956:
5949:
5940:
5939:
5865:
5864:
5781:
5780:
5747:
5746:
5724:Branch and bound
5714:Greedy algorithm
5694:
5693:
5681:
5680:
5601:
5600:
5557:
5556:
5544:
5543:
5485:
5484:
5472:
5471:
5420:Truncated Newton
5335:Wolfe conditions
5318:
5317:
5271:
5270:
5258:
5257:
5231:
5224:
5217:
5208:
5207:
5195:Brian Borchers,
5162:
5148:
5127:
5118:
5072:
5060:
5049:
5018:
4974:
4953:
4951:
4949:
4943:
4931:
4909:
4887:
4860:
4854:
4848:
4845:
4839:
4838:
4836:
4834:
4828:
4821:
4812:
4801:
4800:
4798:
4796:
4790:
4783:
4774:
4759:
4752:
4746:
4743:
4737:
4736:
4734:
4733:
4719:
4713:
4712:
4710:
4698:
4692:
4691:
4689:
4688:
4674:
4668:
4667:
4665:
4663:
4657:
4651:. Archived from
4650:
4641:
4586:
4585:
4556:
4550:
4549:
4513:
4507:
4506:
4488:
4482:
4471:
4465:
4464:
4454:
4436:
4427:
4421:
4420:
4402:
4384:
4375:
4369:
4368:
4358:
4349:
4348:
4341:
4335:
4334:
4332:
4330:
4311:
4300:
4271:
4270:
4250:
4244:
4243:
4223:
4217:
4216:
4192:
4186:
4183:
4177:
4176:
4166:
4138:
4132:
4127:
3801:MATLAB, Octave,
3663:uncertainty sets
3458:
3455:
3425:
3423:
3422:
3417:
3405:
3403:
3402:
3397:
3385:
3383:
3382:
3377:
3365:
3363:
3362:
3357:
3349:
3348:
3332:
3330:
3329:
3324:
3322:
3321:
3303:
3302:
3283:
3281:
3280:
3275:
3263:
3261:
3260:
3255:
3240:
3238:
3237:
3232:
3224:
3223:
3204:
3202:
3201:
3196:
3176:
3175:
3148:
3147:
3128:
3126:
3125:
3120:
3104:
3102:
3101:
3096:
3079:
3078:
3069:
3068:
3041:
3040:
3031:
3030:
3013:
3011:
3010:
3005:
2994:
2993:
2977:
2975:
2974:
2969:
2958:
2957:
2939:
2938:
2926:
2925:
2908:
2906:
2905:
2900:
2879:
2877:
2876:
2871:
2866:
2865:
2847:
2846:
2834:
2833:
2805:
2803:
2802:
2797:
2778:
2776:
2775:
2770:
2765:
2764:
2746:
2745:
2733:
2732:
2716:
2714:
2713:
2708:
2696:
2694:
2693:
2688:
2676:
2674:
2673:
2668:
2656:
2654:
2653:
2648:
2633:
2631:
2630:
2625:
2611:
2610:
2601:
2600:
2573:
2572:
2563:
2562:
2538:
2537:
2522:
2521:
2503:
2502:
2490:
2489:
2455:
2453:
2452:
2447:
2442:
2438:
2422:
2421:
2394:
2393:
2364:
2363:
2347:
2345:
2344:
2339:
2337:
2336:
2323:
2321:
2320:
2315:
2291:
2289:
2288:
2283:
2266:
2265:
2249:
2247:
2246:
2241:
2214:
2207:
2203:
2200:
2194:
2163:
2155:
2124:Ellipsoid method
2079:barrier function
2073:General problems
2067:steepest descent
2063:gradient descent
1859:
1857:
1856:
1851:
1849:
1811:
1810:
1809:
1804:
1795:
1783:
1782:
1772:
1770:
1761:
1737:
1730:
1729:
1728:
1723:
1714:
1698:
1696:
1694:
1683:
1541:
1539:
1538:
1533:
1531:
1499:
1497:
1488:
1464:
1457:
1456:
1446:
1444:
1442:
1431:
1405:
1403:
1402:
1397:
1395:
1354:
1346:
1345:
1335:
1334:
1333:
1295:
1287:
1286:
1276:
1275:
1274:
1255:
1243:
1241:
1232:
1208:
1200:
1198:
1196:
1189:
1177:
1143:
1141:
1140:
1135:
1120:
1118:
1117:
1112:
1100:concave function
1090:
1088:
1087:
1082:
1080:
1079:
1066:
1064:
1063:
1058:
1056:
1055:
1046:
1034:
1032:
1031:
1026:
1010:
1008:
1007:
1002:
1000:
999:
984:is a vector and
983:
981:
980:
975:
973:
972:
971:
954:
952:
951:
946:
944:
943:
931:
923:
922:
921:
905:
897:
896:
876:
874:
873:
868:
838:
836:
835:
830:
828:
820:
819:
814:
805:
804:
785:
783:
782:
777:
747:
745:
744:
739:
737:
729:
728:
723:
714:
713:
690:
688:
687:
682:
680:
672:
671:
666:
657:
656:
634:
632:
631:
626:
624:
623:
618:
609:
591:
589:
588:
583:
581:
540:
532:
531:
521:
520:
519:
481:
473:
472:
462:
460:
451:
427:
420:
408:
406:
404:
393:
362:
360:
359:
354:
335:
333:
332:
327:
315:
313:
312:
307:
273:If such a point
264:
262:
261:
256:
245:
234:
207:
205:
204:
199:
191:
190:
189:
168:
166:
165:
160:
158:
157:
152:
123:
121:
120:
115:
113:
105:
104:
99:
90:
89:
28:convex functions
6415:
6414:
6410:
6409:
6408:
6406:
6405:
6404:
6390:Convex analysis
6380:
6379:
6378:
6373:
6357:
6324:
6279:
6210:
6136:
6127:Semi-continuity
6112:Convex function
6093:Logarithmically
6060:
6021:Convex geometry
6002:
5993:Convex function
5976:
5970:Convex analysis
5967:
5937:
5932:
5915:
5872:
5851:
5814:
5775:
5752:
5741:
5734:
5688:
5663:
5627:
5594:
5585:
5562:
5551:
5530:
5504:
5500:Penalty methods
5495:Barrier methods
5479:
5466:
5446:
5442:Newton's method
5424:
5376:
5339:
5307:
5288:Powell's method
5265:
5252:
5235:
5176:
5145:Convex analysis
5099:
5069:
5038:
5007:
4971:
4947:
4945:
4941:
4928:
4906:
4884:
4868:
4863:
4855:
4851:
4846:
4842:
4832:
4830:
4826:
4819:
4813:
4804:
4794:
4792:
4788:
4781:
4775:
4762:
4753:
4749:
4744:
4740:
4731:
4729:
4721:
4720:
4716:
4699:
4695:
4686:
4684:
4676:
4675:
4671:
4661:
4659:
4655:
4648:
4642:
4589:
4582:
4558:
4557:
4553:
4514:
4510:
4503:
4489:
4485:
4472:
4468:
4461:10.1137/1035044
4452:10.1.1.161.7209
4434:
4428:
4424:
4382:
4376:
4372:
4359:
4352:
4343:
4342:
4338:
4328:
4326:
4324:
4309:
4301:
4274:
4267:
4251:
4247:
4240:
4224:
4220:
4193:
4189:
4184:
4180:
4139:
4135:
4128:
4121:
4117:
4085:
4069:convex analysis
4053:
3963:control systems
3959:
3573:MATLAB, Octave
3432:
3411:
3408:
3407:
3391:
3388:
3387:
3371:
3368:
3367:
3344:
3340:
3338:
3335:
3334:
3317:
3313:
3298:
3294:
3292:
3289:
3288:
3269:
3266:
3265:
3249:
3246:
3245:
3219:
3215:
3213:
3210:
3209:
3171:
3167:
3143:
3139:
3137:
3134:
3133:
3114:
3111:
3110:
3074:
3070:
3064:
3060:
3036:
3032:
3026:
3022:
3020:
3017:
3016:
2989:
2985:
2983:
2980:
2979:
2953:
2949:
2934:
2930:
2921:
2917:
2915:
2912:
2911:
2885:
2882:
2881:
2861:
2857:
2842:
2838:
2829:
2825:
2811:
2808:
2807:
2791:
2788:
2787:
2760:
2756:
2741:
2737:
2728:
2724:
2722:
2719:
2718:
2702:
2699:
2698:
2682:
2679:
2678:
2677:that minimizes
2662:
2659:
2658:
2642:
2639:
2638:
2637:For each point
2606:
2602:
2596:
2592:
2568:
2564:
2558:
2554:
2533:
2529:
2517:
2513:
2498:
2494:
2485:
2481:
2467:
2464:
2463:
2417:
2413:
2389:
2385:
2372:
2368:
2359:
2358:
2356:
2353:
2352:
2332:
2331:
2329:
2326:
2325:
2297:
2294:
2293:
2261:
2257:
2255:
2252:
2251:
2226:
2223:
2222:
2215:
2204:
2198:
2195:
2180:
2164:
2153:
2113:self-concordant
2075:
2052:Newton's method
2026:
2021:
1971:
1877:
1847:
1846:
1805:
1800:
1799:
1791:
1787:
1778:
1774:
1771:
1739:
1735:
1734:
1724:
1719:
1718:
1710:
1706:
1697:
1690:
1685:
1680:
1678:
1675:
1674:
1669:
1622:
1611:
1575:
1568:
1551:linear subspace
1529:
1528:
1498:
1466:
1462:
1461:
1452:
1448:
1445:
1438:
1433:
1428:
1426:
1423:
1422:
1412:
1393:
1392:
1350:
1341:
1337:
1331:
1330:
1291:
1282:
1278:
1272:
1271:
1251:
1242:
1210:
1206:
1205:
1199:
1185:
1184:
1179:
1174:
1172:
1169:
1168:
1158:linear function
1150:
1126:
1123:
1122:
1106:
1103:
1102:
1091:is convex, the
1075:
1074:
1072:
1069:
1068:
1051:
1050:
1042:
1040:
1037:
1036:
1020:
1017:
1016:
995:
991:
989:
986:
985:
967:
963:
962:
960:
957:
956:
939:
935:
927:
917:
913:
912:
901:
892:
888:
886:
883:
882:
844:
841:
840:
824:
815:
810:
809:
800:
796:
794:
791:
790:
753:
750:
749:
733:
724:
719:
718:
709:
705:
703:
700:
699:
693:convex function
676:
667:
662:
661:
652:
651:
643:
640:
639:
619:
614:
613:
605:
603:
600:
599:
579:
578:
536:
527:
523:
517:
516:
477:
468:
464:
461:
429:
425:
424:
416:
407:
400:
395:
390:
388:
385:
384:
374:
348:
345:
344:
321:
318:
317:
301:
298:
297:
241:
230:
216:
213:
212:
185:
181:
180:
178:
175:
174:
153:
148:
147:
139:
136:
135:
109:
100:
95:
94:
85:
84:
76:
73:
72:
65:convex function
53:
48:
17:
12:
11:
5:
6413:
6403:
6402:
6397:
6392:
6375:
6374:
6372:
6371:
6365:
6363:
6359:
6358:
6356:
6355:
6350:
6348:Strong duality
6345:
6340:
6334:
6332:
6326:
6325:
6323:
6322:
6287:
6285:
6281:
6280:
6278:
6277:
6272:
6263:
6258:
6256:John ellipsoid
6253:
6248:
6243:
6238:
6224:
6218:
6216:
6212:
6211:
6209:
6208:
6203:
6198:
6193:
6188:
6183:
6178:
6173:
6168:
6163:
6158:
6153:
6147:
6145:
6143:results (list)
6138:
6137:
6135:
6134:
6129:
6124:
6119:
6117:Invex function
6114:
6105:
6100:
6095:
6090:
6085:
6079:
6074:
6068:
6066:
6062:
6061:
6059:
6058:
6053:
6048:
6043:
6038:
6033:
6028:
6023:
6018:
6016:Choquet theory
6012:
6010:
6004:
6003:
6001:
6000:
5995:
5990:
5984:
5982:
5981:Basic concepts
5978:
5977:
5966:
5965:
5958:
5951:
5943:
5934:
5933:
5931:
5930:
5924:
5921:
5920:
5917:
5916:
5914:
5913:
5908:
5903:
5898:
5893:
5888:
5883:
5877:
5874:
5873:
5870:Metaheuristics
5861:
5860:
5857:
5856:
5853:
5852:
5850:
5849:
5844:
5842:Ford–Fulkerson
5839:
5834:
5828:
5826:
5820:
5819:
5816:
5815:
5813:
5812:
5810:Floyd–Warshall
5807:
5802:
5801:
5800:
5789:
5787:
5777:
5776:
5774:
5773:
5768:
5763:
5757:
5755:
5744:
5736:
5735:
5733:
5732:
5731:
5730:
5716:
5711:
5706:
5700:
5698:
5690:
5689:
5677:
5676:
5673:
5672:
5669:
5668:
5665:
5664:
5662:
5661:
5656:
5651:
5646:
5640:
5638:
5629:
5628:
5626:
5625:
5620:
5615:
5613:Affine scaling
5609:
5607:
5605:Interior point
5598:
5587:
5586:
5584:
5583:
5578:
5573:
5567:
5565:
5553:
5552:
5540:
5539:
5536:
5535:
5532:
5531:
5529:
5528:
5523:
5518:
5512:
5510:
5509:Differentiable
5506:
5505:
5503:
5502:
5497:
5491:
5489:
5481:
5480:
5468:
5467:
5457:
5455:
5452:
5451:
5448:
5447:
5445:
5444:
5438:
5436:
5430:
5429:
5426:
5425:
5423:
5422:
5417:
5412:
5407:
5402:
5397:
5392:
5386:
5384:
5378:
5377:
5375:
5374:
5369:
5364:
5355:
5349:
5347:
5341:
5340:
5338:
5337:
5332:
5326:
5324:
5315:
5309:
5308:
5306:
5305:
5300:
5295:
5290:
5285:
5279:
5277:
5267:
5266:
5254:
5253:
5234:
5233:
5226:
5219:
5211:
5205:
5204:
5199:
5193:
5187:
5175:
5174:External links
5172:
5171:
5170:
5163:
5150:
5149:
5137:
5128:
5119:
5097:
5073:
5067:
5050:
5036:
5019:
5005:
4988:
4976:
4975:
4969:
4954:
4932:
4926:
4910:
4904:
4888:
4882:
4867:
4864:
4862:
4861:
4849:
4840:
4802:
4760:
4747:
4738:
4727:www.cvxgrp.org
4714:
4693:
4669:
4587:
4580:
4551:
4524:(1): 129–171.
4508:
4502:978-0898715156
4501:
4483:
4466:
4445:(2): 183–238.
4422:
4370:
4350:
4336:
4322:
4272:
4265:
4245:
4238:
4218:
4187:
4178:
4149:(2): 117–129.
4133:
4118:
4116:
4113:
4112:
4111:
4106:
4101:
4096:
4091:
4084:
4081:
4052:
4049:
4048:
4047:
4044:
4037:
4031:
4025:
4018:
4011:regularization
4005:Variations of
4003:
4000:
3997:
3971:circuit design
3958:
3955:
3952:
3951:
3949:
3947:
3944:
3942:
3938:
3937:
3935:
3932:
3929:
3927:
3923:
3922:
3920:
3917:
3914:
3912:
3908:
3907:
3905:
3902:
3899:
3897:
3893:
3892:
3890:
3887:
3884:
3882:
3878:
3877:
3875:
3872:
3869:
3867:
3863:
3862:
3860:
3857:
3854:
3852:
3848:
3847:
3845:
3842:
3839:
3836:
3832:
3831:
3829:
3826:
3823:
3818:
3814:
3813:
3811:
3808:
3805:
3799:
3795:
3794:
3792:
3789:
3786:
3784:
3780:
3779:
3777:
3774:
3771:
3768:
3762:
3761:
3759:
3756:
3749:
3744:
3740:
3739:
3737:
3734:
3731:
3729:
3725:
3724:
3722:
3719:
3716:
3714:
3710:
3709:
3707:
3704:
3697:
3695:
3691:
3690:
3688:
3685:
3682:
3676:
3672:
3671:
3669:
3666:
3659:
3657:
3653:
3652:
3650:
3647:
3640:
3638:
3634:
3633:
3631:
3628:
3625:
3622:
3618:
3617:
3615:
3612:
3609:
3607:
3603:
3602:
3600:
3597:
3594:
3591:
3587:
3586:
3584:
3581:
3574:
3571:
3567:
3566:
3564:
3561:
3559:
3554:
3550:
3549:
3547:
3544:
3541:
3536:
3532:
3531:
3529:
3527:
3525:
3522:
3518:
3517:
3515:
3512:
3505:
3500:
3496:
3495:
3493:
3490:
3487:
3482:
3478:
3477:
3474:
3468:
3465:
3462:
3440:modeling tools
3431:
3428:
3415:
3395:
3375:
3355:
3352:
3347:
3343:
3320:
3316:
3312:
3309:
3306:
3301:
3297:
3273:
3253:
3230:
3227:
3222:
3218:
3206:
3205:
3194:
3191:
3188:
3185:
3182:
3179:
3174:
3170:
3166:
3163:
3160:
3157:
3154:
3151:
3146:
3142:
3118:
3107:
3106:
3094:
3091:
3088:
3085:
3082:
3077:
3073:
3067:
3063:
3059:
3056:
3053:
3050:
3047:
3044:
3039:
3035:
3029:
3025:
3014:
3003:
3000:
2997:
2992:
2988:
2967:
2964:
2961:
2956:
2952:
2948:
2945:
2942:
2937:
2933:
2929:
2924:
2920:
2909:
2898:
2895:
2892:
2889:
2869:
2864:
2860:
2856:
2853:
2850:
2845:
2841:
2837:
2832:
2828:
2824:
2821:
2818:
2815:
2795:
2768:
2763:
2759:
2755:
2752:
2749:
2744:
2740:
2736:
2731:
2727:
2706:
2686:
2666:
2646:
2635:
2634:
2623:
2620:
2617:
2614:
2609:
2605:
2599:
2595:
2591:
2588:
2585:
2582:
2579:
2576:
2571:
2567:
2561:
2557:
2553:
2550:
2547:
2544:
2541:
2536:
2532:
2528:
2525:
2520:
2516:
2512:
2509:
2506:
2501:
2497:
2493:
2488:
2484:
2480:
2477:
2474:
2471:
2457:
2456:
2445:
2441:
2437:
2434:
2431:
2428:
2425:
2420:
2416:
2412:
2409:
2406:
2403:
2400:
2397:
2392:
2388:
2384:
2381:
2378:
2375:
2371:
2367:
2362:
2335:
2313:
2310:
2307:
2304:
2301:
2281:
2278:
2275:
2272:
2269:
2264:
2260:
2239:
2236:
2233:
2230:
2217:
2216:
2167:
2165:
2158:
2152:
2149:
2137:
2136:
2131:
2126:
2121:
2116:
2106:
2100:
2097:Bundle methods
2074:
2071:
2045:KKT conditions
2034:linear algebra
2025:
2022:
2020:
2017:
1997:
1996:
1989:
1986:
1983:global minimum
1970:
1967:
1966:
1965:
1959:
1954:
1949:
1940:
1939:
1933:
1927:
1921:
1915:
1876:
1873:
1845:
1842:
1839:
1836:
1833:
1830:
1827:
1823:
1820:
1817:
1814:
1808:
1803:
1798:
1794:
1790:
1786:
1781:
1777:
1773:
1769:
1766:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1738:
1736:
1733:
1727:
1722:
1717:
1713:
1709:
1705:
1702:
1699:
1693:
1689:
1684:
1682:
1667:
1620:
1609:
1573:
1567:
1564:
1543:
1542:
1527:
1524:
1521:
1518:
1515:
1512:
1509:
1506:
1503:
1500:
1496:
1493:
1487:
1484:
1481:
1478:
1475:
1472:
1469:
1465:
1463:
1460:
1455:
1451:
1447:
1441:
1437:
1432:
1430:
1411:
1408:
1407:
1406:
1391:
1388:
1385:
1382:
1379:
1376:
1373:
1370:
1366:
1363:
1360:
1357:
1353:
1349:
1344:
1340:
1336:
1332:
1329:
1326:
1323:
1320:
1317:
1314:
1311:
1307:
1304:
1301:
1298:
1294:
1290:
1285:
1281:
1277:
1273:
1270:
1267:
1264:
1261:
1258:
1254:
1250:
1247:
1244:
1240:
1237:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1209:
1207:
1204:
1201:
1195:
1192:
1188:
1183:
1178:
1176:
1164:, as follows:
1149:
1146:
1133:
1130:
1110:
1078:
1054:
1049:
1045:
1024:
1013:
1012:
998:
994:
970:
966:
942:
938:
934:
930:
926:
920:
916:
911:
908:
904:
900:
895:
891:
866:
863:
860:
857:
854:
851:
848:
827:
823:
818:
813:
808:
803:
799:
787:
775:
772:
769:
766:
763:
760:
757:
736:
732:
727:
722:
717:
712:
708:
696:
679:
675:
670:
665:
660:
655:
650:
647:
636:
622:
617:
612:
608:
593:
592:
577:
574:
571:
568:
565:
562:
559:
556:
552:
549:
546:
543:
539:
535:
530:
526:
522:
518:
515:
512:
509:
506:
503:
500:
497:
493:
490:
487:
484:
480:
476:
471:
467:
463:
459:
456:
450:
447:
444:
441:
438:
435:
432:
428:
426:
423:
419:
415:
412:
409:
403:
399:
394:
392:
373:
370:
369:
368:
352:
343:Otherwise, if
341:
325:
305:
294:
267:
266:
254:
251:
248:
244:
240:
237:
233:
229:
226:
223:
220:
197:
194:
188:
184:
171:
170:
156:
151:
146:
143:
125:
112:
108:
103:
98:
93:
88:
83:
80:
52:
49:
47:
44:
15:
9:
6:
4:
3:
2:
6412:
6401:
6398:
6396:
6393:
6391:
6388:
6387:
6385:
6370:
6367:
6366:
6364:
6360:
6354:
6351:
6349:
6346:
6344:
6341:
6339:
6336:
6335:
6333:
6331:
6327:
6320:
6318:
6312:
6310:
6304:
6300:
6296:
6292:
6289:
6288:
6286:
6282:
6276:
6273:
6271:
6267:
6264:
6262:
6259:
6257:
6254:
6252:
6249:
6247:
6244:
6242:
6239:
6237:
6233:
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6213:
6207:
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6199:
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6187:
6186:Mazur's lemma
6184:
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6174:
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6132:Subderivative
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6008:Topics (list)
6005:
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5912:
5909:
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5904:
5902:
5899:
5897:
5894:
5892:
5889:
5887:
5886:Hill climbing
5884:
5882:
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5878:
5875:
5871:
5866:
5862:
5848:
5845:
5843:
5840:
5838:
5835:
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5830:
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5827:
5825:
5824:Network flows
5821:
5811:
5808:
5806:
5803:
5799:
5796:
5795:
5794:
5791:
5790:
5788:
5786:
5785:Shortest path
5782:
5772:
5769:
5767:
5764:
5762:
5759:
5758:
5756:
5754:
5753:spanning tree
5748:
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5707:
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5695:
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5686:Combinatorial
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5382:Other methods
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4682:www.cvxpy.org
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4658:on 2017-09-18
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4061:pseudo-convex
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4030:optimization.
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2168:This section
2166:
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2146:
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2048:
2046:
2042:
2037:
2035:
2031:
2030:unconstrained
2016:
2014:
2013:Farkas' lemma
2010:
2006:
2002:
1994:
1990:
1987:
1984:
1980:
1979:local minimum
1976:
1975:
1974:
1963:
1960:
1958:
1955:
1953:
1950:
1948:
1947:Least squares
1945:
1944:
1943:
1937:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1913:
1910:
1909:
1905:
1901:
1897:
1893:
1889:
1884:
1880:
1875:Special cases
1872:
1870:
1866:
1860:
1843:
1840:
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1828:
1825:
1821:
1818:
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1165:
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1159:
1155:
1145:
1131:
1128:
1108:
1101:
1096:
1094:
1093:sublevel sets
1047:
1022:
996:
992:
940:
936:
932:
924:
909:
893:
889:
880:
864:
861:
858:
855:
852:
849:
846:
816:
806:
801:
797:
788:
773:
770:
767:
764:
761:
758:
755:
725:
715:
710:
706:
697:
694:
668:
658:
648:
645:
637:
620:
610:
598:
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596:
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572:
569:
566:
563:
560:
557:
554:
550:
547:
544:
528:
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513:
510:
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504:
501:
498:
495:
491:
488:
485:
469:
465:
410:
397:
383:
382:
381:
379:
378:standard form
372:Standard form
366:
350:
342:
339:
323:
303:
295:
292:
288:
284:
280:
279:optimal point
276:
272:
271:
270:
249:
246:
238:
224:
211:
210:
209:
195:
192:
186:
154:
144:
141:
134:
133:convex subset
131:, which is a
130:
126:
101:
91:
81:
78:
70:
66:
62:
58:
57:
56:
51:Abstract form
43:
41:
37:
33:
29:
25:
21:
6353:Weak duality
6316:
6308:
6228:Orthogonally
6030:
5891:Local search
5837:Edmonds–Karp
5793:Bellman–Ford
5563:minimization
5548:
5395:Gauss–Newton
5345:Quasi–Newton
5330:Trust region
5238:Optimization
5166:
5158:
5144:
5131:
5123:
5080:
5056:
5027:
4996:
4984:
4959:
4946:. Retrieved
4937:
4917:
4895:
4873:
4852:
4843:
4831:. Retrieved
4793:. Retrieved
4755:
4750:
4741:
4730:. Retrieved
4726:
4717:
4696:
4685:. Retrieved
4681:
4672:
4660:. Retrieved
4653:the original
4563:
4554:
4521:
4517:
4511:
4492:
4486:
4469:
4442:
4438:
4425:
4393:(1): 42–60.
4390:
4386:
4373:
4363:
4339:
4327:. Retrieved
4305:
4255:
4248:
4228:
4221:
4204:
4200:
4190:
4181:
4164:2027.42/6740
4146:
4142:
4136:
4054:
3960:
3957:Applications
3851:ConicBundle
3679:
3467:Description
3452:
3448:
3443:
3439:
3435:
3433:
3243:
3207:
3108:
2636:
2458:
2220:
2205:
2196:
2181:Please help
2169:
2141:dual problem
2138:
2091:
2086:
2076:
2060:
2049:
2038:
2029:
2027:
1998:
1992:
1972:
1941:
1878:
1868:
1864:
1862:
1673:
1664:
1660:
1656:
1652:
1648:
1644:
1640:
1636:
1632:
1628:
1624:
1617:
1613:
1606:
1602:
1598:
1597:columns. If
1594:
1590:
1586:
1582:
1578:
1571:
1569:
1560:
1557:
1554:
1544:
1415:
1413:
1153:
1151:
1097:
1014:
1011:is a scalar.
594:
377:
375:
364:
337:
290:
286:
282:
278:
274:
268:
172:
129:feasible set
128:
71:variables,
68:
60:
54:
19:
18:
6343:Duality gap
6338:Dual system
6222:Convex hull
5911:Tabu search
5322:Convergence
5293:Line search
4857:Ahmad Bazzi
4475:Ruszczyński
4439:SIAM Review
4065:quasiconvex
4041:uncertainty
4009:(including
3129:satisfying
2056:line search
287:optimal set
32:convex sets
6384:Categories
6266:Radial set
6236:Convex set
5998:Convex set
5742:algorithms
5250:heuristics
5242:Algorithms
4983:. (2004).
4944:. Springer
4866:References
4732:2021-06-17
4687:2021-04-12
4400:1709.04494
4051:Extensions
3979:statistics
3713:SparsePOP
3535:Convex.jl
3444:interfaces
2806:minimizes
2199:April 2021
2019:Algorithms
1969:Properties
1416:conic form
1410:Conic form
1162:constraint
365:infeasible
208:attaining
46:Definition
6251:Hypograph
5697:Paradigms
5596:quadratic
5313:Gradients
5275:Functions
4708:1410.4821
4538:0025-5610
4479:Bertsekas
4447:CiteSeerX
4207:: 15–22.
3694:SOSTOOLS
3464:Language
3342:λ
3315:λ
3308:…
3296:λ
3217:λ
3162:…
3062:λ
3055:⋯
3024:λ
2987:λ
2960:≥
2951:λ
2944:…
2932:λ
2919:λ
2891:∈
2880:over all
2859:λ
2852:…
2840:λ
2827:λ
2758:λ
2751:…
2739:λ
2726:λ
2594:λ
2587:⋯
2556:λ
2531:λ
2515:λ
2508:…
2496:λ
2483:λ
2433:≤
2408:…
2377:∈
2309:≤
2303:≤
2277:≤
2170:does not
2041:quadratic
1838:…
1816:≤
1549:, L is a
1523:∩
1505:∈
1381:…
1322:…
1300:≤
1266:≤
1260:−
1129:−
1048:∈
933:−
925:⋅
859:…
822:→
768:…
731:→
674:→
659:⊆
611:∈
567:…
508:…
486:≤
338:unbounded
247:∈
193:∈
187:∗
145:⊆
107:→
92:⊆
6275:Zonotope
6246:Epigraph
5928:Software
5805:Dijkstra
5636:exchange
5434:Hessians
5400:Gradient
5157:(2006).
5143:(1970).
5026:(1993).
4995:(1993).
4916:(2015).
4894:(2009).
4824:Archived
4786:Archived
4566:. 2006.
4546:28882966
4417:67856259
4173:30500771
4083:See also
4057:biconvex
3678:MATLAB,
3590:LMI lab
3511:solver.
3461:Program
3430:Software
1993:strictly
1688:minimize
1623:, where
1589:, where
1436:minimize
1182:minimize
955:, where
398:minimize
291:solvable
283:solution
6330:Duality
6232:Pseudo-
6206:Ursescu
6103:Pseudo-
6077:Concave
6056:Simplex
6036:Duality
5771:Kruskal
5761:BorĹŻvka
5751:Minimum
5488:General
5246:methods
5126:. SIAM.
5115:9048698
5107:1900016
5046:1295240
5015:1261420
4089:Duality
3985:), and
3975:finance
3896:PENNON
3798:SeDuMi
3770:Python
3680:Octave
3624:MATLAB
3593:MATLAB
3570:YALMIP
3524:Python
3499:CVXMOD
3436:solvers
3286:scalars
2779:called
2191:removed
2176:sources
2143:. The
2087:phase I
1898:, SDP:
1894:, SOCP
1643:), and
1581:)=0 as
595:where:
40:NP-hard
6313:, and
6284:Series
6201:Simons
6108:Quasi-
6098:Proper
6083:Closed
5633:Basis-
5591:Linear
5561:Convex
5405:Mirror
5362:L-BFGS
5248:, and
5113:
5105:
5095:
5065:
5044:
5034:
5013:
5003:
4967:
4948:12 Apr
4924:
4902:
4880:
4833:12 Apr
4795:12 Apr
4662:12 Apr
4578:
4544:
4536:
4499:
4449:
4415:
4329:12 Apr
4320:
4263:
4236:
4171:
4063:, and
3911:SDPLR
3835:SDPT3
3783:MOSEK
3766:CVXOPT
3728:CPLEX
3637:AIMMS
3521:CVXPY
3509:CVXOPT
3503:Python
3485:MATLAB
2011:, and
2007:, the
1977:every
1902:, CP:
1890:, QP:
1762:
1647:is an
1639:-rank(
1627:is in
1489:
1233:
877:, are
452:
6141:Main
5832:Dinic
5740:Graph
5111:S2CID
4942:(PDF)
4827:(PDF)
4820:(PDF)
4789:(PDF)
4782:(PDF)
4703:arXiv
4656:(PDF)
4649:(PDF)
4542:S2CID
4435:(PDF)
4413:S2CID
4395:arXiv
4383:(PDF)
4310:(PDF)
4169:S2CID
4115:Notes
3926:GAMS
3881:LOQO
3866:DSDP
3817:SDPA
3743:CSDP
3656:ROME
3621:xLMI
3553:CVXR
3539:Julia
3406:over
3366:then
3333:with
2697:over
1981:is a
1156:is a
691:is a
30:over
6261:Lens
6215:Sets
6065:Maps
5972:and
5798:SPFA
5766:Prim
5360:and
5182:and
5093:ISBN
5063:ISBN
5032:ISBN
5001:ISBN
4965:ISBN
4950:2021
4922:ISBN
4900:ISBN
4878:ISBN
4835:2021
4797:2021
4664:2021
4576:ISBN
4534:ISSN
4497:ISBN
4331:2021
4318:ISBN
4261:ISBN
4234:ISBN
4013:and
3919:Yes
3874:Yes
3859:Yes
3844:Yes
3828:Yes
3810:Yes
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3630:Yes
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3492:Yes
3481:CVX
3476:Ref
3471:FOSS
3442:(or
3187:<
2996:>
2348:is:
2292:for
2174:any
2172:cite
1651:-by-
1593:has
1553:of R
127:The
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6315:(Hw
5728:cut
5593:and
5085:doi
4568:doi
4526:doi
4457:doi
4405:doi
4209:doi
4159:hdl
4151:doi
3934:No
3904:No
3889:No
3821:C++
3803:MEX
3791:No
3736:No
3649:No
3599:No
3264:in
2657:in
2185:by
2069:).
296:If
281:or
219:inf
67:of
6386::
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