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arxiv: 2604.20992 · v1 · submitted 2026-04-22 · 🧮 math.OC

Optimization Workshop Notes for Mathematical Programming with Equilibrium Constraints (MPECs): Second-Order Optimality Conditions

Jiguang Yu This is my paper

Pith reviewed 2026-05-09 23:38 UTC · model grok-4.3

classification 🧮 math.OC
keywords MPECsecond-order optimality conditionsequilibrium constraintsmultiplier-based conditionsimplicit programmingpiecewise programmingcritical conesstrong regularity
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The pith

Second-order optimality conditions for MPECs require three viewpoints to handle equilibrium constraints where classical nonlinear programming fails.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper delivers a compact introduction to second-order optimality conditions for mathematical programs with equilibrium constraints. It starts from the classical nonlinear programming template and explains its failure in the MPEC context due to the lower-level equilibrium system. The notes then develop the three main viewpoints in the literature: multiplier-based conditions, implicit-programming conditions based on the solution map, and piecewise-programming conditions from decomposing the complementarity structure. The focus remains on conceptual structure, critical cones, strong regularity, and curvature terms. A sympathetic reader would care as these provide practical ways to verify local optimality in optimization problems common to economics, engineering, and competitive systems.

Core claim

By moving beyond the classical nonlinear programming template, second-order optimality conditions for MPECs are obtained via multiplier-based conditions, implicit-programming conditions that use the solution map of the lower-level equilibrium system, and piecewise-programming conditions that decompose the complementarity into smooth pieces, with the argument centering on critical cones and strong regularity to incorporate curvature terms correctly.

What carries the argument

The critical cone and strong regularity of the equilibrium system, which enable the construction of the three types of second-order conditions.

If this is right

  • Multiplier-based conditions extend standard second-order necessary and sufficient conditions by including additional multipliers and curvature information for the equilibrium constraints.
  • Implicit-programming conditions derive optimality by considering the derivative of the implicit solution map of the lower-level problem.
  • Piecewise-programming conditions allow local analysis by considering each smooth piece of the decomposed complementarity system as a standard nonlinear program.
  • These conditions together address the degeneracy that occurs when the lower-level problem has multiple solutions or active constraints at equilibrium.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The notes may enable direct application to bilevel optimization problems by viewing them as special cases of MPECs.
  • Extensions could include numerical algorithms that check these conditions without full analytical derivation of the cones.
  • Connections to sensitivity analysis in parametric variational inequalities could strengthen the implicit-programming viewpoint.
  • Testable extensions involve applying the conditions to specific MPEC instances from traffic or energy models to validate local optimality predictions.

Load-bearing premise

The lower-level equilibrium system is strongly regular or admits a piecewise smooth structure that allows the three viewpoints to apply without additional qualifications.

What would settle it

An MPEC where a point satisfies the second-order conditions from one or more viewpoints but is not a local minimizer, or fails the conditions yet is optimal.

Figures

Figures reproduced from arXiv: 2604.20992 by Jiguang Yu.

Figure 1
Figure 1. Figure 1: Local picture: the feasible set of an MPEC is typically branchwise smooth rather [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Tangent cone and critical cone at a boundary stationary point. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Why second-order MPEC theory departs from standard NLP theory. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Local union-of-polyhedra geometry in the AVI case. [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Why β(¯z) is the technically difficult set in NCP-constrained MPECs. 5.3 Multiplier-based second-order condition Because the lower-level data are nonlinear, curvature terms of the NCP mapping appear. For￾mally, one expects a quadratic form of type dzT [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Schematic local graph of the implicit solution map [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Interpretation of y (2)(¯x; dx) as second-order deviation. 6.4 Second-order conditions in reduced form The reduced second-order expression is dzT∇2 f(¯z) dz + ∇yf(¯z) T y (2)(¯x; dx), dz = (dx, y′ (¯x; dx)). Theorem 6.3 (Implicit-programming necessary condition). Assume z¯ = (¯x, y¯) is a stationary point of (4), the lower-level NCP is strongly regular at z¯, and f, F are C 2 near z¯. If z¯ is a local mini… view at source ↗
Figure 8
Figure 8. Figure 8: The refined multiplier formula is obtained by eliminating [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Variable roles in the KKT-constrained MPEC formulation. [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Piecewise-programming viewpoint: solve second-order analysis branch by branch. [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: A good pedagogical sequencing for reading the notes. [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
read the original abstract

In this workshop, we present a compact but rigorous introduction to second-order optimality conditions for mathematical programs with equilibrium constraints (MPECs). We start from the classical nonlinear programming template, then explain why it fails in the equilibrium-constrained setting, and develop the three main viewpoints used in the literature: (i) multiplier-based conditions, (ii) implicit-programming conditions based on the solution map of the lower-level equilibrium system, and (iii) piecewise-programming conditions obtained by decomposing complementarity structure into smooth pieces. The emphasis is on conceptual structure, critical cones, strong regularity, and the exact role of curvature terms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript consists of workshop notes offering a compact introduction to second-order optimality conditions for mathematical programs with equilibrium constraints (MPECs). It begins with the standard nonlinear programming (NLP) template, highlights its shortcomings in the MPEC context, and elaborates on three primary approaches from the literature: (i) multiplier-based conditions, (ii) implicit-programming conditions relying on the solution map of the lower-level equilibrium, and (iii) piecewise-programming conditions derived from decomposing the complementarity structure into smooth pieces. The focus is on key concepts such as critical cones, strong regularity, and the role of curvature terms.

Significance. If the presentation is accurate and clear, these notes could provide a useful structured overview for understanding the different perspectives on second-order conditions in MPECs, drawing from established literature. As purely expository material without novel results or derivations, the significance lies in its potential pedagogical value for workshops or self-study rather than in advancing theoretical knowledge.

minor comments (2)
  1. The abstract provides a good overview, but it would benefit from explicitly stating the target audience's assumed background in nonlinear programming and variational inequalities to align with the weakest assumption noted.
  2. Ensure that all references to cited literature are complete and that any diagrams or examples illustrating the three viewpoints are clearly labeled and explained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation of minor revision. The manuscript is intended as compact workshop notes providing a structured introduction to second-order optimality conditions for MPECs, drawing on established literature without claiming novel results. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

Expository notes with no new derivations or claims

full rationale

The manuscript consists of workshop notes that organize and explain three established viewpoints on second-order optimality conditions for MPECs drawn from the existing literature (multiplier-based, implicit-programming via solution maps, and piecewise decomposition). No new theorems, derivations, quantitative predictions, or fitted quantities are asserted. The content reduces directly to standard concepts in nonlinear programming, variational inequalities, and complementarity problems as cited, with no internal derivation chain that could reduce to its own inputs by construction. All load-bearing elements are external to the notes themselves.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is an expository document; no free parameters, new axioms, or invented entities are introduced beyond standard background assumptions in optimization theory.

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Forward citations

Cited by 3 Pith papers

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Reference graph

Works this paper leans on

46 extracted references · 46 canonical work pages · cited by 3 Pith papers

  1. [1]

    Local structure of feasible sets in nonlinear programming, part i: Regularity

    Stephen M Robinson. Local structure of feasible sets in nonlinear programming, part i: Regularity. InNumerical Methods: Proceedings of the International Workshop Held at Caracas, June 14–18, 1982, pages 240–251. Springer, 2007

    work page 1982

  2. [2]

    Local structure of feasible sets in nonlinear programming, part ii: Nondegeneracy

    Stephen M Robinson. Local structure of feasible sets in nonlinear programming, part ii: Nondegeneracy. InMathematical Programming at Oberwolfach II, pages 217–230. Springer, 2009

    work page 2009

  3. [3]

    Local structure of feasible sets in nonlinear programming, part iii: Stability and sensitivity

    Stephen M Robinson. Local structure of feasible sets in nonlinear programming, part iii: Stability and sensitivity. InNonlinear Analysis and Optimization, pages 45–66. Springer, 2009

    work page 2009

  4. [4]

    Bidirectional endothelial feedback drives turing-vascular patterning and drug-resistance niches: a hybrid pde-agent-based study.Bioengineering, 12(10):1097, 2025

    Zonghao Liu, Louis Shuo Wang, Jiguang Yu, Jilin Zhang, Erica Martel, and Shijia Li. Bidirectional endothelial feedback drives turing-vascular patterning and drug-resistance niches: a hybrid pde-agent-based study.Bioengineering, 12(10):1097, 2025

    work page 2025

  5. [5]

    Projected gradient descent accumulates at bouli- gand stationary points.SIAM Journal on Optimization, 35(2):1004–1029, 2025

    Guillaume Olikier and Ir` ene Waldspurger. Projected gradient descent accumulates at bouli- gand stationary points.SIAM Journal on Optimization, 35(2):1004–1029, 2025

    work page 2025

  6. [6]

    Lipschitz stability of least- squares problems regularized by functions with c 2-cone reducible conjugates.Mathematics of Operations Research, 2026

    Ying Cui, Tim Hoheisel, Tran TA Nghia, and Defeng Sun. Lipschitz stability of least- squares problems regularized by functions with c 2-cone reducible conjugates.Mathematics of Operations Research, 2026

    work page 2026

  7. [7]

    Monotone inclusions, acceleration, and closed-loop con- trol.Mathematics of Operations Research, 48(4):2353–2382, 2023

    Tianyi Lin and Michael I Jordan. Monotone inclusions, acceleration, and closed-loop con- trol.Mathematics of Operations Research, 48(4):2353–2382, 2023

    work page 2023

  8. [8]

    On the b-differential of the componentwise minimum of two affine vector functions: J.-p

    Jean-Pierre Dussault, Jean Charles Gilbert, and Baptiste Plaquevent-Jourdain. On the b-differential of the componentwise minimum of two affine vector functions: J.-p. dussault et al.Mathematical Programming Computation, 17(1):1–52, 2025

    work page 2025

  9. [9]

    Analysis framework for stochastic predator–prey model with demographic noise.Results in Applied Mathematics, 27:100621, 2025

    Louis Shuo Wang and Jiguang Yu. Analysis framework for stochastic predator–prey model with demographic noise.Results in Applied Mathematics, 27:100621, 2025

    work page 2025

  10. [10]

    Generalized polyhedral dc optimization problems: V

    Vu Thi Huong, Duong Thi Kim Huyen, and Nguyen Dong Yen. Generalized polyhedral dc optimization problems: V. huong et al.Journal of Optimization Theory and Applications, 207(1):11, 2025

    work page 2025

  11. [11]

    Generalized equations and their solutions, part ii: Applications to nonlinear programming

    Stephen M Robinson. Generalized equations and their solutions, part ii: Applications to nonlinear programming. InOptimality and stability in mathematical programming, pages 200–221. Springer, 2009

    work page 2009

  12. [12]

    SIAM, 2021

    Alexander Shapiro, Darinka Dentcheva, and Andrzej Ruszczynski.Lectures on stochastic programming: modeling and theory. SIAM, 2021

    work page 2021

  13. [13]

    A globally con- vergent proximal newton-type method in nonsmooth convex optimization.Mathematical Programming, 198(1):899–936, 2023

    Boris S Mordukhovich, Xiaoming Yuan, Shangzhi Zeng, and Jin Zhang. A globally con- vergent proximal newton-type method in nonsmooth convex optimization.Mathematical Programming, 198(1):899–936, 2023

    work page 2023

  14. [14]

    Flexible differentiable optimization via model transformations.INFORMS Journal on Computing, 36(2):456–478, 2024

    Mathieu Besan¸ con, Joaquim Dias Garcia, Benoˆ ıt Legat, and Akshay Sharma. Flexible differentiable optimization via model transformations.INFORMS Journal on Computing, 36(2):456–478, 2024

    work page 2024

  15. [15]

    Algebraic–spectral thresholds and discrete–continuous stability transfer in leslie–gower systems.Electronic Research Archive, 34(1):251–290, 2026

    Louis Shuo Wang and Jiguang Yu. Algebraic–spectral thresholds and discrete–continuous stability transfer in leslie–gower systems.Electronic Research Archive, 34(1):251–290, 2026. 19

    work page 2026

  16. [16]

    On optimality conditions for nonlinear conic programming.Mathematics of Operations Research, 47(3):2160–2185, 2022

    Roberto Andreani, Walter G´ omez, Gabriel Haeser, Leonardo M Mito, and Alberto Ramos. On optimality conditions for nonlinear conic programming.Mathematics of Operations Research, 47(3):2160–2185, 2022

    work page 2022

  17. [17]

    An exact penalization viewpoint of constrained optimization.SIAM Journal on control and optimization, 29(4):968–998, 1991

    James V Burke. An exact penalization viewpoint of constrained optimization.SIAM Journal on control and optimization, 29(4):968–998, 1991

    work page 1991

  18. [18]

    Constrained consensus-based op- timization.SIAM Journal on Optimization, 33(1):211–236, 2023

    Giacomo Borghi, Michael Herty, and Lorenzo Pareschi. Constrained consensus-based op- timization.SIAM Journal on Optimization, 33(1):211–236, 2023

    work page 2023

  19. [19]

    A sequential quadratic programming algorithm for nonsmooth problems with upper-objective.SIAM Journal on Optimization, 33(3):2379– 2405, 2023

    Jingyi Wang and Cosmin G Petra. A sequential quadratic programming algorithm for nonsmooth problems with upper-objective.SIAM Journal on Optimization, 33(3):2379– 2405, 2023

    work page 2023

  20. [20]

    Local search for nonsmooth dc optimization with dc equality and inequality constraints

    Alexander S Strekalovsky. Local search for nonsmooth dc optimization with dc equality and inequality constraints. InNumerical nonsmooth optimization: state of the art algorithms, pages 229–261. Springer, 2020

    work page 2020

  21. [21]

    Sufficient conditions for metric subregu- larity of constraint systems with applications to disjunctive and ortho-disjunctive programs

    Mat´ uˇ s Benko, MichalˇCervinka, and Tim Hoheisel. Sufficient conditions for metric subregu- larity of constraint systems with applications to disjunctive and ortho-disjunctive programs. Set-Valued and Variational Analysis, 30(1):143–177, 2022

    work page 2022

  22. [22]

    Global well-posedness and stability of nonlocal damage-structured lineage model with feedback and dedifferentiation

    Ye Liang, Louis Shuo Wang, Jiguang Yu, and Zonghao Liu. Global well-posedness and stability of nonlocal damage-structured lineage model with feedback and dedifferentiation. Mathematics, 13(22):3583, 2025

    work page 2025

  23. [23]

    Exact augmented lagrangian duality for mixed integer quadratic programming.SIAM Journal on Optimization, 30(1):781–797, 2020

    Xiaoyi Gu, Shabbir Ahmed, and Santanu S Dey. Exact augmented lagrangian duality for mixed integer quadratic programming.SIAM Journal on Optimization, 30(1):781–797, 2020

    work page 2020

  24. [24]

    John Wiley & Sons, Inc., 1983

    Garth P McCormick.Nonlinear programming: Theory, algorithms, and applications. John Wiley & Sons, Inc., 1983

    work page 1983

  25. [25]

    A branch-and-cut algorithm for mixed-integer bi- linear programming.European Journal of Operational Research, 282(2):506–514, 2020

    Matteo Fischetti and Michele Monaci. A branch-and-cut algorithm for mixed-integer bi- linear programming.European Journal of Operational Research, 282(2):506–514, 2020

    work page 2020

  26. [26]

    Exact and ap- proximate schemes for robust optimization problems with decision-dependent information discovery.INFORMS Journal on Computing, 37(6):1457–1477, 2025

    Rosario Paradiso, Angelos Georghiou, Said Dabia, and Denise T¨ onissen. Exact and ap- proximate schemes for robust optimization problems with decision-dependent information discovery.INFORMS Journal on Computing, 37(6):1457–1477, 2025

    work page 2025

  27. [27]

    Strongly regular generalized equations.Mathematics of Operations Research, 5(1):43–62, 1980

    Stephen M Robinson. Strongly regular generalized equations.Mathematics of Operations Research, 5(1):43–62, 1980

    work page 1980

  28. [28]

    Analysis and mean-field limit of a hybrid PDE-ABM modeling angiogenesis-regulated resistance evolution.Mathematics, 13(17):2898, 2025

    Louis Shuo Wang, Jiguang Yu, Shijia Li, and Zonghao Liu. Analysis and mean-field limit of a hybrid PDE-ABM modeling angiogenesis-regulated resistance evolution.Mathematics, 13(17):2898, 2025

    work page 2025

  29. [29]

    Near- optimal distributed linear-quadratic regulator for networked systems.SIAM Journal on Control and Optimization, 61(3):1113–1135, 2023

    Sungho Shin, Yiheng Lin, Guannan Qu, Adam Wierman, and Mihai Anitescu. Near- optimal distributed linear-quadratic regulator for networked systems.SIAM Journal on Control and Optimization, 61(3):1113–1135, 2023

    work page 2023

  30. [30]

    Exponential decay of sensitivity in graph-structured nonlinear programs.SIAM Journal on Optimization, 32(2):1156–1183, 2022

    Sungho Shin, Mihai Anitescu, and Victor M Zavala. Exponential decay of sensitivity in graph-structured nonlinear programs.SIAM Journal on Optimization, 32(2):1156–1183, 2022

    work page 2022

  31. [31]

    Springer, 2021

    Asen L Dontchev et al.Lectures on variational analysis, volume 205. Springer, 2021. 20

    work page 2021

  32. [32]

    Differentiating nonsmooth solutions to parametric monotone inclusion problems.SIAM Journal on Optimization, 34 (1):71–97, 2024

    J´ erˆ ome Bolte, Edouard Pauwels, and Antonio Silveti-Falls. Differentiating nonsmooth solutions to parametric monotone inclusion problems.SIAM Journal on Optimization, 34 (1):71–97, 2024

    work page 2024

  33. [33]

    A damage-structured pde model of stem cell hierarchies: The dual role of dedifferentiation in tissue homeostasis and aging.Plos one, 21(2):e0335163, 2026

    Louis Shuo Wang, Jiguang Yu, and Zonghao Liu. A damage-structured pde model of stem cell hierarchies: The dual role of dedifferentiation in tissue homeostasis and aging.Plos one, 21(2):e0335163, 2026

    work page 2026

  34. [34]

    On (local) analysis of multifunctions via subspaces contained in graphs of generalized derivatives.Journal of Mathematical Analysis and Ap- plications, 508(2):125895, 2022

    Helmut Gfrerer and Jiˇ r´ ı V Outrata. On (local) analysis of multifunctions via subspaces contained in graphs of generalized derivatives.Journal of Mathematical Analysis and Ap- plications, 508(2):125895, 2022

    work page 2022

  35. [35]

    Geometric characterizations of lipschitz stability for convex optimization problems.SIAM Journal on Optimization, 35(2):927–958, 2025

    Tran TA Nghia. Geometric characterizations of lipschitz stability for convex optimization problems.SIAM Journal on Optimization, 35(2):927–958, 2025

    work page 2025

  36. [36]

    Aubin property and strong regularity are equivalent for nonlinear second-order cone programming.SIAM Journal on Optimization, 35(2):712–738, 2025

    Liang Chen, Ruoning Chen, Defeng Sun, and Junyuan Zhu. Aubin property and strong regularity are equivalent for nonlinear second-order cone programming.SIAM Journal on Optimization, 35(2):712–738, 2025

    work page 2025

  37. [37]

    Strong variational suf- ficiency for nonlinear semidefinite programming and its implications.SIAM Journal on Optimization, 33(4):2988–3011, 2023

    Shiwei Wang, Chao Ding, Yangjing Zhang, and Xinyuan Zhao. Strong variational suf- ficiency for nonlinear semidefinite programming and its implications.SIAM Journal on Optimization, 33(4):2988–3011, 2023

    work page 2023

  38. [38]

    Newton’s method for b-differentiable equations.Mathematics of operations research, 15(2):311–341, 1990

    Jong-Shi Pang. Newton’s method for b-differentiable equations.Mathematics of operations research, 15(2):311–341, 1990

    work page 1990

  39. [39]

    Glob- ally convergent coderivative-based generalized newton methods in nonsmooth optimization

    Pham Duy Khanh, Boris S Mordukhovich, Vo Thanh Phat, and Dat Ba Tran. Glob- ally convergent coderivative-based generalized newton methods in nonsmooth optimization. Mathematical Programming, 205(1):373–429, 2024

    work page 2024

  40. [40]

    Jiguang Yu, Louis Shuo Wang, Zonghao Liu, and Jingfeng Liu. Pattern suppression and recovery under one-way versus two-way chemotactic coupling in hybrid partial differential equation–ordinary differential equation models.Transport Phenomena, (0), 2026

    work page 2026

  41. [41]

    Exact computation of an error bound for the balanced linear complementarity problem with unique solution.Mathematical Pro- gramming, 199(1):1221–1238, 2023

    Jean-Pierre Dussault and Jean Charles Gilbert. Exact computation of an error bound for the balanced linear complementarity problem with unique solution.Mathematical Pro- gramming, 199(1):1221–1238, 2023

    work page 2023

  42. [42]

    A generalized newton method for subgradient systems.Mathematics of Operations Research, 48(4):1811–1845, 2023

    Pham Duy Khanh, Boris Mordukhovich, and Vo Thanh Phat. A generalized newton method for subgradient systems.Mathematics of Operations Research, 48(4):1811–1845, 2023

    work page 2023

  43. [43]

    Solving two-stage stochastic variational inequalities by a hybrid projection semismooth newton algorithm.SIAM Journal on Scientific Computing, 45(4):A1741–A1765, 2023

    Xiaozhou Wang and Xiaojun Chen. Solving two-stage stochastic variational inequalities by a hybrid projection semismooth newton algorithm.SIAM Journal on Scientific Computing, 45(4):A1741–A1765, 2023

    work page 2023

  44. [44]

    Lecture note for bounded controls in continuous-time and control of several variables.arXiv e-prints, pages arXiv–2604, 2026

    Louis Shuo Wang. Lecture note for bounded controls in continuous-time and control of several variables.arXiv e-prints, pages arXiv–2604, 2026

    work page 2026

  45. [45]

    Polyhedral newton-min algorithms for complementarity problems: J.-p

    Jean-Pierre Dussault, Mathieu Frappier, and Jean Charles Gilbert. Polyhedral newton-min algorithms for complementarity problems: J.-p. dussault et al.Mathematical Programming, 215(1):269–324, 2026

    work page 2026

  46. [46]

    Sensitivity analysis in nonlinear programs and variational inequalities via continuous selections.SIAM Journal on Control and Optimization, 33(4):1040–1060, 1995

    Jiming Liu. Sensitivity analysis in nonlinear programs and variational inequalities via continuous selections.SIAM Journal on Control and Optimization, 33(4):1040–1060, 1995. 21

    work page 1995