Automatic repair of convex optimization problems
Given an infeasible, unbounded, or pathological convex optimization problem, a natural question to ask is: what is the smallest change we can make to the problem’s parameters such that the problem becomes solvable? In this paper, we address this question by posing it as an optimization problem invol...
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| Published in | Optimization and engineering Vol. 22; no. 1; pp. 247 - 259 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.03.2021
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 1389-4420 1573-2924 |
| DOI | 10.1007/s11081-020-09508-9 |
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| Abstract | Given an infeasible, unbounded, or pathological convex optimization problem, a natural question to ask is: what is the smallest change we can make to the problem’s parameters such that the problem becomes solvable? In this paper, we address this question by posing it as an optimization problem involving the minimization of a convex regularization function of the parameters, subject to the constraint that the parameters result in a solvable problem. We propose a heuristic for approximately solving this problem that is based on the penalty method and leverages recently developed methods that can efficiently evaluate the derivative of the solution of a convex cone program with respect to its parameters. We illustrate our method by applying it to examples in optimal control and economics. |
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| AbstractList | Given an infeasible, unbounded, or pathological convex optimization problem, a natural question to ask is: what is the smallest change we can make to the problem’s parameters such that the problem becomes solvable? In this paper, we address this question by posing it as an optimization problem involving the minimization of a convex regularization function of the parameters, subject to the constraint that the parameters result in a solvable problem. We propose a heuristic for approximately solving this problem that is based on the penalty method and leverages recently developed methods that can efficiently evaluate the derivative of the solution of a convex cone program with respect to its parameters. We illustrate our method by applying it to examples in optimal control and economics. |
| Author | Angeris, Guillermo Barratt, Shane Boyd, Stephen |
| Author_xml | – sequence: 1 givenname: Shane surname: Barratt fullname: Barratt, Shane email: stbarratt@gmail.com organization: Department of Electrical Engineering, Stanford University – sequence: 2 givenname: Guillermo surname: Angeris fullname: Angeris, Guillermo organization: Department of Electrical Engineering, Stanford University – sequence: 3 givenname: Stephen surname: Boyd fullname: Boyd, Stephen organization: Department of Electrical Engineering, Stanford University |
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| Keywords | Parametric optimization Unboundedness Convex programming Infeasibility |
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Ph.D. thesis Ben-Tal A, Nemirovski A (2001) Lectures on modern convex optimization: analysis, algorithms, and engineering applications, vol 2. SIAM Amaldi E, Pfetsch M, Trotter L (1999) Some structural and algorithmic properties of the maximum feasible subsystem problem. In: International conference on integer programming and combinatorial optimization. Springer, pp 45–59 GUROBI Optimization (2019) Gurobi optimizer reference manual KellnerKPfetschMTheobaldTIrreducible infeasible subsystems of semidefinite systemsJ Optim Theory Appl20191813727742395183710.1007/s10957-019-01480-4 KuratorWO’NeillRPERUSE: an interactive system for mathematical programsACM Trans Math Softw19806448950910.1145/355921.355923 ParikhNBoydSProximal algorithmsFound Trends® Optim20141312723910.1561/2400000003 Agrawal A, Amos B, Barratt S, Boyd S, Diamond S, Kolter J.Z (2019a) Differentiable convex optimization layers. 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Springer, pp 79–97 – reference: O’DonoghueBChuEParikhNBoydSConic optimization via operator splitting and homogeneous self-dual embeddingJ Optim Theory Appl2016169310421068350139710.1007/s10957-016-0892-3 – reference: MOSEK Aps (2020) MOSEK optimizer API for Python. https://docs.mosek.com – reference: ObuchowskaWInfeasibility analysis for systems of quadratic convex inequalitiesEur J Oper Res1998107363364310.1016/S0377-2217(97)00168-9 – reference: ObuchowskaWOn infeasibility of systems of convex analytic inequalitiesJ Math Anal Appl19992341223245169483710.1006/jmaa.1999.6357 – reference: ChinneckJAnalyzing mathematical programs using MProbeAnn Oper Res20011041–43348187751610.1023/A:1013178600790 – reference: KellnerKPfetschMTheobaldTIrreducible infeasible subsystems of semidefinite systemsJ Optim Theory Appl20191813727742395183710.1007/s10957-019-01480-4 – reference: GreenbergHMurphyFApproaches to diagnosing infeasible linear programsORSA J Comput19913325326110.1287/ijoc.3.3.253 – reference: PfetschMBranch-and-cut for the maximum feasible subsystem problemSIAM J Optim20081912138240302010.1137/050645828 – reference: TamizMMardleSJonesDDetecting IIS in infeasible linear programmes using techniques from goal programmingComput Oper Res199623211311910.1016/0305-0548(95)00018-H – reference: Amaldi E (1994) From finding maximum feasible subsystems of linear systems to feedforward neural network design. Ph.D. thesis, Citeseer – reference: DiamondSBoydSCVXPY: a Python-embedded modeling language for convex optimizationJ Mach Learn Res201617831534829211360.90008 – reference: GreenbergHANALYZE: a computer-assisted analysis system for linear programming modelsOper Res Lett19876524925510.1016/0167-6377(87)90057-5 – reference: Barratt S, Boyd S (2019) Least squares auto-tuning. arXiv preprint arXiv:1904.05460 – reference: ChinneckJFeasibility and infeasibility in optimization: algorithms and computational methods2007BerlinSpringer1178.90369 – reference: Pfetsch M (2003) The maximum feasible subsystem problem and vertex-facet incidences of polyhedra. Ph.D. thesis – reference: Ben-Tal A, Nemirovski A (2001) Lectures on modern convex optimization: analysis, algorithms, and engineering applications, vol 2. SIAM – reference: ChinneckJAn effective polynomial-time heuristic for the minimum-cardinality IIS set-covering problemAnn Math Artif Intell1996171127144141573010.1007/BF02284627 – reference: GUROBI Optimization (2019) Gurobi optimizer reference manual – reference: IBM (2016) IBM ILOG CPLEX optimization studio CPLEX user’s manual – reference: Van LoonJNMIrreducibly inconsistent systems of linear inequalitiesEur J Oper Res19818328328863740110.1016/0377-2217(81)90177-6 – reference: Agrawal A, Amos B, Barratt S, Boyd S, Diamond S, Kolter J.Z (2019a) Differentiable convex optimization layers. In: Advances in neural information processing systems, pp 9558–9570 – reference: AgrawalABarrattSBoydSBussetiEMoursiWDifferentiating through a cone programJ Appl Numer Optim201912107115 – reference: Amaldi E, Pfetsch M, Trotter L (1999) Some structural and algorithmic properties of the maximum feasible subsystem problem. In: International conference on integer programming and combinatorial optimization. Springer, pp 45–59 – reference: Karp R (1972) Reducibility among combinatorial problems. In: Complexity of computer computations, pp 85–103 – reference: ChinneckJFinding a useful subset of constraints for analysis in an infeasible linear programINFORMS J Comput199792164174147731210.1287/ijoc.9.2.164 – reference: RoodmanGNote–post-infeasibility analysis in linear programmingManage Sci197925991692256096510.1287/mnsc.25.9.916 – reference: KuratorWO’NeillRPERUSE: an interactive system for mathematical programsACM Trans Math Softw19806448950910.1145/355921.355923 – reference: AmaralPJúdiceJSheraliHA reformulation–linearization–convexification algorithm for optimal correction of an inconsistent system of linear constraintsComput Oper Res200835514941509257832110.1016/j.cor.2006.08.007 – reference: ParikhNBoydSProximal algorithmsFound Trends® Optim20141312723910.1561/2400000003 – reference: MartinetBBrève communication. régularisation d’inéquations variationnelles par approximations successivesESAIM: Mathematical Modelling and Numerical Analysis-Modélisation Mathématique et Analyse Numérique19704R31541580215.21103 – reference: Carver W (1922) Systems of linear inequalities. Ann Math 212–220 – reference: Gambella C, Marecek J, Mevissen M (2019) Projections onto the set of feasible inputs and the set of feasible solutions. In: Allerton conference on communication, control, and computing. 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| Title | Automatic repair of convex optimization problems |
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