pith. sign in

arxiv: 2605.00386 · v1 · submitted 2026-05-01 · 🧮 math.OC

Introduction to Mathematical Programming with Equilibrium Constraints (MPECs) and Bilevel Optimization

Louis Shuo Wang This is my paper

Pith reviewed 2026-05-09 19:31 UTC · model grok-4.3

classification 🧮 math.OC
keywords MPECequilibrium constraintsbilevel optimizationvariational inequalitiescomplementarity problemsmathematical programmingexistence theoryoptimization applications
0
0 comments X

The pith

An MPEC is an optimization problem whose feasible set is partly defined by another optimization, variational inequality, complementarity system, or equilibrium model.

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

This paper introduces mathematical programs with equilibrium constraints as optimization problems where part of what counts as feasible comes from solving a separate equilibrium or lower-level optimization. It motivates the structure by pointing to applications in which one decision maker's choices depend on the optimal response of another. The work presents the main ways to rewrite those equilibrium conditions in equivalent forms that fit standard mathematical programming. It also summarizes basic conditions under which solutions to the overall problem are guaranteed to exist. A reader cares because this pattern appears whenever decisions are layered or interdependent through optimality conditions.

Core claim

The central message is that an MPEC is an optimization problem whose feasible set is partly defined by another optimization, variational inequality, complementarity system, or equilibrium model.

What carries the argument

The equilibrium constraint, which requires that a subset of the variables satisfy the optimality or equilibrium conditions of a separate model.

If this is right

  • Bilevel optimization problems become special cases of MPECs once the lower-level optimality is expressed as an equilibrium constraint.
  • Equivalent reformulations allow the use of existing nonlinear programming algorithms on the resulting single-level problem.
  • Existence results give verifiable conditions on the data that guarantee at least one optimal solution exists.
  • Applications such as market clearing or network flows can be cast directly as MPECs and analyzed with the summarized theory.

Where Pith is reading between the lines

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

  • The same structure could be used to model multi-period decisions where each period's equilibrium depends on the previous period's solution.
  • Testing the reformulations on a small traffic equilibrium instance would show whether computational savings appear in practice.
  • Connections to game-theoretic settings with multiple interacting equilibria remain open for further development.

Load-bearing premise

Readers already possess sufficient background in mathematical optimization to understand the equivalent formulations and existence theory being summarized.

What would settle it

A concrete example of an equilibrium model whose conditions cannot be rewritten in any of the listed equivalent forms would falsify the claim that those reformulations cover the main cases.

read the original abstract

Our aim is to explain mathematical programs with equilibrium constraints (MPECs), motivate them through applications, present the main equivalent formulations of equilibrium constraints, and summarize the basic existence theory for optimal solutions. The central message is that an MPEC is an optimization problem whose feasible set is partly defined by another optimization, variational inequality, complementarity system, or equilibrium model.

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 is an expository introduction to mathematical programs with equilibrium constraints (MPECs) and bilevel optimization. It defines an MPEC as an optimization problem whose feasible set is partly defined by an embedded equilibrium model (another optimization problem, variational inequality, complementarity system, or equilibrium model). The paper motivates the topic via applications, presents main equivalent formulations of the equilibrium constraints, and summarizes basic existence theory for optimal solutions.

Significance. As a purely expository survey that consolidates standard definitions, reformulations, and existence results from the established MPEC literature, the paper has moderate significance. It may serve as a useful entry point for readers with prior optimization background by organizing accessible explanations of equivalent formulations and basic theory. No new derivations, parameter-free results, or machine-checked proofs are claimed or provided.

minor comments (2)
  1. [Abstract] The abstract states the central message clearly but does not indicate the assumed background level or the paper's organizational structure (e.g., section outline).
  2. Notation for the embedded equilibrium models (e.g., how the lower-level problem is denoted) should be introduced with a dedicated preliminary section to aid readers new to the area.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the positive recommendation to accept. The referee's summary correctly reflects the paper's expository goals of motivating MPECs through applications, presenting equivalent formulations of equilibrium constraints, and summarizing basic existence results.

Circularity Check

0 steps flagged

No significant circularity; expository survey of standard definitions

full rationale

This manuscript is a purely expository introduction that restates the established definition of an MPEC as an optimization problem whose feasible set is constrained by an embedded equilibrium model (optimization, VI, or complementarity system). It summarizes equivalent reformulations and basic existence results from the literature without any new derivations, parameter fits, predictions, or claims of novelty. No load-bearing steps reduce by construction to the paper's own inputs, and the central message requires no internal proof or self-referential construction. The work is self-contained against external benchmarks as a survey, consistent with the default non-circular outcome for such texts.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is an expository introduction with no new mathematical claims, derivations, or postulated entities.

pith-pipeline@v0.9.0 · 5340 in / 937 out tokens · 21352 ms · 2026-05-09T19:31:11.724473+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. A Diagnostic Framework for Implementation Risk in Bilevel Decision Problems: The Ambiguity Premium and the Robustness--Efficiency Frontier

    math.OC 2026-05 unverdicted novelty 6.0

    The paper defines the ambiguity premium Δ_ε(x) as the gap between pessimistic and optimistic upper-level values over ε-optimal follower responses and provides bounds plus a screening workflow to trace robustness-effic...

Reference graph

Works this paper leans on

103 extracted references · 103 canonical work pages · cited by 1 Pith paper · 3 internal anchors

  1. [1]

    Mathematical Programming70(1), 47–72 (1995)

    Falk, J.E., Liu, J.: On bilevel programming, part i: general nonlinear cases. Mathematical Programming70(1), 47–72 (1995)

    work page 1995

  2. [2]

    Journal of Optimization Theory and Applications205(2), 31 (2025) 18

    Aghasi, A., Ghadimi, S.: Fully zeroth-order bilevel programming via gaussian smoothing. Journal of Optimization Theory and Applications205(2), 31 (2025) 18

    work page 2025

  3. [3]

    SIAM Journal on Optimization33(1), 147–180 (2023)

    Hong, M., Wai, H.-T., Wang, Z., Yang, Z.: A two-timescale stochastic algo- rithm framework for bilevel optimization: Complexity analysis and application to actor-critic. SIAM Journal on Optimization33(1), 147–180 (2023)

    work page 2023

  4. [4]

    Numerical Functional Analysis and Optimization 15(7-8), 869–887 (1994)

    Koˇ ccvara, M., Outrata, J.V.: On optimization of systems governed by implicit complementarity problems. Numerical Functional Analysis and Optimization 15(7-8), 869–887 (1994)

    work page 1994

  5. [5]

    SIAM Journal on Control and Optimization58(6), 3237–3261 (2020)

    Chaudet-Dumas, B., Deteix, J.: Shape derivatives for the penalty formulation of elastic contact problems with tresca friction. SIAM Journal on Control and Optimization58(6), 3237–3261 (2020)

    work page 2020

  6. [6]

    In: 2023 62nd IEEE Conference on Decision and Control (CDC), pp

    Liu, S., Xiao, W., Belta, C.A.: Auxiliary-variable adaptive control barrier func- tions for safety critical systems. In: 2023 62nd IEEE Conference on Decision and Control (CDC), pp. 8602–8607 (2023). IEEE

    work page 2023

  7. [7]

    Bioengineering12(10), 1097 (2025)

    Liu, Z., Wang, L.S., Yu, J., Zhang, J., Martel, E., Li, S.: Bidirectional endothelial feedback drives turing-vascular patterning and drug-resistance niches: a hybrid pde-agent-based study. Bioengineering12(10), 1097 (2025)

    work page 2025

  8. [8]

    Mathematical Programming68(1), 105–130 (1995)

    Outrata, J., Zowe, J.: A numerical approach to optimization problems with variational inequality constraints. Mathematical Programming68(1), 105–130 (1995)

    work page 1995

  9. [9]

    Mathematical Programming 198(2), 1153–1225 (2023)

    Cui, S., Shanbhag, U.V., Yousefian, F.: Complexity guarantees for an implicit smoothing-enabled method for stochastic mpecs. Mathematical Programming 198(2), 1153–1225 (2023)

    work page 2023

  10. [10]

    SIAM Journal on Optimization30(3), 2163–2196 (2020)

    Christof, C., Reyes, J.C., Meyer, C.: A nonsmooth trust-region method for locally lipschitz functions with application to optimization problems constrained by variational inequalities. SIAM Journal on Optimization30(3), 2163–2196 (2020)

    work page 2020

  11. [11]

    Journal of Optimization Theory and Applications209(1), 15 (2026)

    Rawat, A., Singh, V.: Augmented lagrangian neural network for solving mathe- matical programs with equilibrium constraints. Journal of Optimization Theory and Applications209(1), 15 (2026)

    work page 2026

  12. [12]

    Mathematics of Opera- tions Research5(1), 43–62 (1980)

    Robinson, S.M.: Strongly regular generalized equations. Mathematics of Opera- tions Research5(1), 43–62 (1980)

    work page 1980

  13. [13]

    In: 2025 IEEE 64th Conference on Decision and Control (CDC), pp

    Chen, Y.-H., Liu, S., Xiao, W., Belta, C., Otte, M.: Control barrier functions via minkowski operations for safe navigation among polytopic sets. In: 2025 IEEE 64th Conference on Decision and Control (CDC), pp. 4481–4488 (2025). IEEE

    work page 2025

  14. [14]

    SIAM Journal on Control and Optimization61(3), 1113–1135 (2023) 19

    Shin, S., Lin, Y., Qu, G., Wierman, A., Anitescu, M.: Near-optimal distributed linear-quadratic regulator for networked systems. SIAM Journal on Control and Optimization61(3), 1113–1135 (2023) 19

    work page 2023

  15. [15]

    SIAM Journal on Optimization34(1), 71–97 (2024)

    Bolte, J., Pauwels, E., Silveti-Falls, A.: Differentiating nonsmooth solutions to parametric monotone inclusion problems. SIAM Journal on Optimization34(1), 71–97 (2024)

    work page 2024

  16. [16]

    SIAM Journal on Optimization35(2), 927–958 (2025)

    Nghia, T.T.: Geometric characterizations of lipschitz stability for convex opti- mization problems. SIAM Journal on Optimization35(2), 927–958 (2025)

    work page 2025

  17. [17]

    Results in Applied Mathematics27, 100621 (2025)

    Wang, L.S., Yu, J.: Analysis framework for stochastic predator–prey model with demographic noise. Results in Applied Mathematics27, 100621 (2025)

    work page 2025

  18. [18]

    SIAM Journal on Optimization35(2), 712–738 (2025)

    Chen, L., Chen, R., Sun, D., Zhu, J.: Aubin property and strong regularity are equivalent for nonlinear second-order cone programming. SIAM Journal on Optimization35(2), 712–738 (2025)

    work page 2025

  19. [19]

    In: 2024 IEEE 63rd Conference on Decision and Control (CDC), pp

    Liu, S., Xiao, W., Belta, C.A.: Auxiliary-variable adaptive control lyapunov barrier functions for spatio-temporally constrained safety-critical applications. In: 2024 IEEE 63rd Conference on Decision and Control (CDC), pp. 8098–8104 (2024). IEEE

    work page 2024

  20. [20]

    In: Analysis and Computation of Fixed Points, pp

    Kojima, M.: Strongly stable stationary solutions in nonlinear programs. In: Analysis and Computation of Fixed Points, pp. 93–138. Elsevier, ??? (1980)

    work page 1980

  21. [21]

    chen et al

    Chen, L., Chen, R., Sun, D., Zhang, L.: Characterizations of the aubin property of the solution mapping for nonlinear semidefinite programming: L. chen et al. Mathematical Programming215(1), 637–668 (2026)

    work page 2026

  22. [22]

    Mathematics of Operations Research (2026)

    Cui, Y., Hoheisel, T., Nghia, T.T., Sun, D.: Lipschitz stability of least- squares problems regularized by functions with c 2-cone reducible conjugates. Mathematics of Operations Research (2026)

    work page 2026

  23. [23]

    SIAM Journal on Optimization32(2), 1156–1183 (2022)

    Shin, S., Anitescu, M., Zavala, V.M.: Exponential decay of sensitivity in graph- structured nonlinear programs. SIAM Journal on Optimization32(2), 1156–1183 (2022)

    work page 2022

  24. [24]

    Journal of Optimization Theory and Applications196(1), 56–77 (2023)

    De Marchi, A., Dreves, A., Gerdts, M., Gottschalk, S., Rogovs, S.: A function approximation approach for parametric optimization. Journal of Optimization Theory and Applications196(1), 56–77 (2023)

    work page 2023

  25. [25]

    Klatte, B

    Bank, B., Guddat, J.: D. Klatte, B. Kummer, and K. Tammer: Non-Linear Parametric Optimization (1982)

    work page 1982

  26. [26]

    Electronic Research Archive34(1), 251–290 (2026)

    Wang, L.S., Yu, J.: Algebraic–spectral thresholds and discrete–continuous stabil- ity transfer in leslie–gower systems. Electronic Research Archive34(1), 251–290 (2026)

    work page 2026

  27. [27]

    Applied Mathematics and Optimization29(2), 20 161–186 (1994)

    Bonnans, J.F.: Local analysis of newton-type methods for variational inequali- ties and nonlinear programming. Applied Mathematics and Optimization29(2), 20 161–186 (1994)

    work page 1994

  28. [28]

    Mathematical Programming205(1), 373–429 (2024)

    Khanh, P.D., Mordukhovich, B.S., Phat, V.T., Tran, D.B.: Globally conver- gent coderivative-based generalized newton methods in nonsmooth optimization. Mathematical Programming205(1), 373–429 (2024)

    work page 2024

  29. [29]

    Mathematics of Operations Research (2025)

    Chen, L., Sun, D., Zhang, W.: Two typical implementable semismooth* new- ton methods for generalized equations are g-semismooth newton methods. Mathematics of Operations Research (2025)

    work page 2025

  30. [30]

    Mathematics of Operations Research47(1), 397–426 (2022)

    Mohammadi, A., Mordukhovich, B.S., Sarabi, M.E.: Variational analysis of com- posite models with applications to continuous optimization. Mathematics of Operations Research47(1), 397–426 (2022)

    work page 2022

  31. [31]

    dussault et al

    Dussault, J.-P., Frappier, M., Gilbert, J.C.: Polyhedral newton-min algorithms for complementarity problems: J.-p. dussault et al. Mathematical Programming 215(1), 269–324 (2026)

    work page 2026

  32. [32]

    Mathematics of Operations Research19(4), 831–879 (1994)

    Gowda, M.S., Pang, J.-S.: Stability analysis of variational inequalities and non- linear complementarity problems, via the mixed linear complementarity problem and degree theory. Mathematics of Operations Research19(4), 831–879 (1994)

    work page 1994

  33. [33]

    IEEE Open Journal of Control Systems (2025)

    Liu, S., Huang, Z., Zeng, J., Sreenath, K., Belta, C.A.: Learning-enabled itera- tive convex optimization for safety-critical model predictive control. IEEE Open Journal of Control Systems (2025)

    work page 2025

  34. [34]

    SIAM Journal on Optimization10(4), 963–981 (2000)

    Qi, L., Wei, Z.: On the constant positive linear dependence condition and its application to sqp methods. SIAM Journal on Optimization10(4), 963–981 (2000)

    work page 2000

  35. [35]

    SIAM Journal on Optimization9(1), 14–32 (1998)

    Facchinei, F., Fischer, A., Kanzow, C.: On the accurate identification of active constraints. SIAM Journal on Optimization9(1), 14–32 (1998)

    work page 1998

  36. [36]

    In: Sensitivity, Stability and Parametric Analysis, pp

    Jittorntrum, K.: Solution point differentiability without strict complementarity in nonlinear programming. In: Sensitivity, Stability and Parametric Analysis, pp. 127–138. Springer, ??? (2009)

    work page 2009

  37. [37]

    Mathematics of Operations Research17(2), 341–364 (1992)

    Kyparisis, J.: Parametric variational inequalities with multivalued solution sets. Mathematics of Operations Research17(2), 341–364 (1992)

    work page 1992

  38. [38]

    SIAM Journal on Control and Optimization33(4), 1040–1060 (1995)

    Liu, J.: Sensitivity analysis in nonlinear programs and variational inequalities via continuous selections. SIAM Journal on Control and Optimization33(4), 1040–1060 (1995)

    work page 1995

  39. [39]

    Recent Advances in Nonsmooth Optimization, 261 (1995) 21

    Pang 1, J.-S.: Stability of parametric nonsmooth equations. Recent Advances in Nonsmooth Optimization, 261 (1995) 21

    work page 1995

  40. [40]

    In: 2025 IEEE 64th Conference on Decision and Control (CDC), pp

    Liu, S., Belta, C.A.: Risk-aware adaptive control barrier functions for safe con- trol of nonlinear systems under stochastic uncertainty. In: 2025 IEEE 64th Conference on Decision and Control (CDC), pp. 4875–4881 (2025). IEEE

    work page 2025

  41. [41]

    In: 2024 IEEE 63rd Conference on Decision and Control (CDC), pp

    Liu, S., Xiao, W., Belta, C.: Feasibility-guaranteed safety-critical control with applications to heterogeneous platoons. In: 2024 IEEE 63rd Conference on Decision and Control (CDC), pp. 8066–8073 (2024). IEEE

    work page 2024

  42. [42]

    Math- ematics of Operations Research17(1), 61–76 (1992)

    Qiu, Y., Magnanti, T.L.: Sensitivity analysis for variational inequalities. Math- ematics of Operations Research17(1), 61–76 (1992)

    work page 1992

  43. [43]

    Journal of Optimization Theory and Applications203(2), 1531–1563 (2024)

    Aussel, D., Chaipunya, P.: Variational and quasi-variational inequalities under local reproducibility: solution concept and applications. Journal of Optimization Theory and Applications203(2), 1531–1563 (2024)

    work page 2024

  44. [44]

    Mathematics of operations research10(1), 54–62 (1985)

    Reinoza, A.: The strong positivity conditions. Mathematics of operations research10(1), 54–62 (1985)

    work page 1985

  45. [45]

    Springer, ??? (2003)

    Facchinei, F., Pang, J.-S.: Finite-dimensional Variational Inequalities and Com- plementarity Problems. Springer, ??? (2003)

    work page 2003

  46. [46]

    Mathematical programming48(1), 161–220 (1990)

    Harker, P.T., Pang, J.-S.: Finite-dimensional variational inequality and nonlin- ear complementarity problems: a survey of theory, algorithms and applications. Mathematical programming48(1), 161–220 (1990)

    work page 1990

  47. [47]

    Mathematics of operations research15(2), 286–298 (1990)

    Kyparisis, J.: Sensitivity analysis for nonlinear programs and variational inequal- ities with nonunique multipliers. Mathematics of operations research15(2), 286–298 (1990)

    work page 1990

  48. [48]

    Information Technology and Control54(2), 413–438 (2025)

    Wang, Z., Wang, D., Yu, J.: Multi-strategy hybrid improved intelligent algo- rithm for solving uav-mtsp. Information Technology and Control54(2), 413–438 (2025)

    work page 2025

  49. [49]

    XIV+ 210 S

    Kleinmichel, H.: AV Fiacco/GP McCormick, Nonlinear Programming: Sequen- tial Unconstrained Minimization Techniques. XIV+ 210 S. m. Fig. New York/London/Sydney/Toronto 1969. John Wiley and Sons, Inc. Preis. geb.£5,

    work page 1969

  50. [50]

    Wiley Online Library (1972)

    work page 1972

  51. [51]

    SIAM, ??? (2000)

    Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. SIAM, ??? (2000)

    work page 2000

  52. [52]

    Mathematics13(17), 2898 (2025)

    Wang, L.S., Yu, J., Li, S., Liu, Z.: Analysis and mean-field limit of a hybrid pde- abm modeling angiogenesis-regulated resistance evolution. Mathematics13(17), 2898 (2025)

    work page 2025

  53. [53]

    SIAM, ??? (2021)

    Bai, Z.-Z., Pan, J.-Y.: Matrix Analysis and Computations. SIAM, ??? (2021)

    work page 2021

  54. [54]

    IEEE Transactions on Vehicular Technology (2026)

    Lin, Z., Lan, J., Liu, S., Tian, Z., Zhao, D., Wei, C.: Hierarchical multi-agent 22 mcts for safety-critical coordination in mixed-autonomy roundabouts. IEEE Transactions on Vehicular Technology (2026)

    work page 2026

  55. [55]

    In: 2022 International Conference on Data Analytics, Computing and Artificial Intelligence (ICDACAI), pp

    Gao, Y., Li, L., Yu, J.: Rolling prediction model of closing price based on eemd data noise reduction and hgs-delm. In: 2022 International Conference on Data Analytics, Computing and Artificial Intelligence (ICDACAI), pp. 255–260 (2022). IEEE

    work page 2022

  56. [56]

    SIAM Review68(1), 3–90 (2026)

    Gander, M.J., Henry, P., Wanner, G.: Landmarks in the history of iterative methods. SIAM Review68(1), 3–90 (2026)

    work page 2026

  57. [57]

    SIAM Journal on Optimization33(3), 1440–1462 (2023)

    Mishchenko, K.: Regularized newton method with global convergence. SIAM Journal on Optimization33(3), 1440–1462 (2023)

    work page 2023

  58. [58]

    SIAM Journal on Optimization34(1), 27–56 (2024)

    Doikov, N., Mishchenko, K., Nesterov, Y.: Super-universal regularized newton method. SIAM Journal on Optimization34(1), 27–56 (2024)

    work page 2024

  59. [59]

    INFORMS Journal on Computing (2025)

    Han, Q., Li, C., Lin, Z., Chen, C., Deng, Q., Ge, D., Liu, H., Ye, Y.: A low-rank admm splitting approach for semidefinite programming. INFORMS Journal on Computing (2025)

    work page 2025

  60. [60]

    Set-Valued and Variational Analysis31(1), 2 (2023)

    Ning, N., Wu, J.: Multi-dimensional path-dependent forward-backward stochas- tic variational inequalities. Set-Valued and Variational Analysis31(1), 2 (2023)

    work page 2023

  61. [61]

    Journal of Optimization Theory and Applications 192(2), 510–532 (2022)

    Wang, J., Ouyang, W.: Newton’s method for solving generalized equations without lipschitz condition. Journal of Optimization Theory and Applications 192(2), 510–532 (2022)

    work page 2022

  62. [62]

    In: Point-to-Set Maps and Mathematical Programming, pp

    Robinson, S.M.: Generalized equations and their solutions, part i: Basic theory. In: Point-to-Set Maps and Mathematical Programming, pp. 128–141. Springer, ??? (2009)

    work page 2009

  63. [63]

    Mathematical Programming198(1), 899–936 (2023)

    Mordukhovich, B.S., Yuan, X., Zeng, S., Zhang, J.: A globally convergent prox- imal newton-type method in nonsmooth convex optimization. Mathematical Programming198(1), 899–936 (2023)

    work page 2023

  64. [64]

    Plos one21(2), 0335163 (2026)

    Wang, L.S., Yu, J., Liu, Z.: A damage-structured pde model of stem cell hier- archies: The dual role of dedifferentiation in tissue homeostasis and aging. Plos one21(2), 0335163 (2026)

    work page 2026

  65. [65]

    In: Nonlinear Analysis and Optimization, pp

    Robinson, S.M.: Local structure of feasible sets in nonlinear programming, part iii: Stability and sensitivity. In: Nonlinear Analysis and Optimization, pp. 45–66. Springer, ??? (2009)

    work page 2009

  66. [66]

    arXiv e-prints, 2604 (2026)

    Shuo Wang, L.: Lecture note for bounded controls in continuous-time and control of several variables. arXiv e-prints, 2604 (2026)

    work page 2026

  67. [67]

    23 Mathematics of Operations Research48(4), 2353–2382 (2023)

    Lin, T., Jordan, M.I.: Monotone inclusions, acceleration, and closed-loop control. 23 Mathematics of Operations Research48(4), 2353–2382 (2023)

    work page 2023

  68. [68]

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

    Yu, J.: Optimization workshop notes for mathematical programming with equi- librium constraints (mpecs): Second-order optimality conditions. arXiv preprint arXiv:2604.20992 (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  69. [69]

    Mathematics of Operations Research50(4), 2873–2908 (2025)

    Liang, L., Sun, D., Toh, K.-C.: A squared smoothing newton method for semidef- inite programming. Mathematics of Operations Research50(4), 2873–2908 (2025)

    work page 2025

  70. [70]

    Linear Algebra and its Applications95, 97–109 (1987)

    Jongen, H.T., M¨ obert, T., R¨ uckmann, J., Tammer, K.: On inertia and schur complement in optimization. Linear Algebra and its Applications95, 97–109 (1987)

    work page 1987

  71. [71]

    Springer, ??? (1990)

    Guddat, J., Vazquez, F.G., Jongen, H.T.: Parametric Optimization: Singulari- ties, Pathfollowing and Jumps. Springer, ??? (1990)

    work page 1990

  72. [72]

    SIAM Journal on Optimization34(1), 1–26 (2024)

    Bellon, A., Dressler, M., Kungurtsev, V., Mareˇ cek, J., Uschmajew, A.: Time- varying semidefinite programming: Path following a burer–monteiro factoriza- tion. SIAM Journal on Optimization34(1), 1–26 (2024)

    work page 2024

  73. [73]

    SIAM Journal on Control and Optimization60(4), 1970–1990 (2022)

    Tang, Y., Dall’Anese, E., Bernstein, A., Low, S.: Running primal-dual gradient method for time-varying nonconvex problems. SIAM Journal on Control and Optimization60(4), 1970–1990 (2022)

    work page 1970

  74. [74]

    SIAM Journal on Optimization35(3), 1524–1550 (2025)

    Liu, H., Grigas, P.: New methods for parametric optimization via differential equations. SIAM Journal on Optimization35(3), 1524–1550 (2025)

    work page 2025

  75. [75]

    Technical report (1979)

    Josephy, N.H.: Newton’s method for generalized equations. Technical report (1979)

    work page 1979

  76. [76]

    SIAM Journal on Optimization34(1), 654–681 (2024)

    Si, W., Absil, P.-A., Huang, W., Jiang, R., Vary, S.: A riemannian proximal newton method. SIAM Journal on Optimization34(1), 654–681 (2024)

    work page 2024

  77. [77]

    Optimization Workshop Notes for Mathematical Programming with Equilibrium Constraints (MPECs): Verification of MPEC Hypotheses

    Yu, J.: Optimization workshop notes for mathematical programming with equi- librium constraints (mpecs): Verification of mpec hypotheses. arXiv preprint arXiv:2604.20988 (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  78. [78]

    arXiv preprint arXiv:2307.15912 (2023)

    Longman, R.W., Liu, S., Elsharhawy, T.A.: A method to speed up convergence of iterative learning control for high precision repetitive motions. arXiv preprint arXiv:2307.15912 (2023)

    work page arXiv 2023

  79. [79]

    SIAM Journal on Optimization33(3), 2021–2040 (2023)

    Yao, W., Yang, X.: Relative lipschitz-like property of parametric systems via projectional coderivatives. SIAM Journal on Optimization33(3), 2021–2040 (2023)

    work page 2021

  80. [80]

    SIAM Journal on Optimization34(1), 870–892 (2024) 24

    Han, S., Pang, J.-S.: Continuous selections of solutions to parametric variational inequalities. SIAM Journal on Optimization34(1), 870–892 (2024) 24

    work page 2024

Showing first 80 references.