MOPECs: Optimization with Equilibrium Constraints

• MOPECs are mathematical models where multiple agents optimize their objectives while satisfying shared equilibrium constraints via variational or complementarity formulations.
• They are widely applied in networked systems, capturing the interplay between strategic upper-level decisions and responsive lower-level equilibria in fields like transportation and economics.
• Solution methods such as nested decomposition, EMP frameworks, and Moment-SOS relaxations enable efficient solving of these complex, coupled optimization problems.

A Multiple Optimization Problem with Equilibrium Constraints (MOPEC) is a class of problems where multiple strategic decision-makers (often called "leaders" or "players") each solve an optimization problem, but their feasible sets and/or objectives are implicitly coupled via the solutions to one or more equilibrium (often variational inequality, complementarity, or Nash equilibrium) problems parameterized by the leaders' own decisions. This structure models hierarchical, multi-agent, and networked systems where upper-level decision-making anticipates or responds to the endogenous equilibria induced in lower-level systems. MOPECs unify and generalize frameworks such as Generalized Nash Equilibrium Problems, multi-leader/follower Stackelberg games, and many classes of economic, engineering, and transportation models.

1. Formal MOPEC Structure and Variational Representations

A MOPEC consists of $N$ optimization agents, each choosing variables $x_i$ to minimize $f_i(x_i, x_{-i})$ over a (possibly coupled) feasible set $K_i(x_{-i})$, and one (or several) equilibrium agents whose feasible decisions are solutions to a parameterized equilibrium problem, e.g., a variational inequality (VI), Nash equilibrium, or vector equilibrium constraint. The canonical formulation is: $$\begin{aligned} &x_i^*\in\arg\min_{x_i\in K_i(x^*_{-i})} f_i(x_i, x^*_{-i}),\quad \text{for }i=1,...,N \ &x_{N+1}^*\in\mathrm{SOL}(K_{N+1}(x^*_{-(N+1)}), F(\cdot, x^*_{-(N+1)})) \end{aligned}$$ where $\mathrm{SOL}(K, F)$ denotes the solution set of $$\langle F(y^*), y-y^* \rangle \ge 0,\, \forall y\in K$$ (Kim et al., 2018).

MOPECs often admit variational and complementarity system representations. Under standard regularity, the first-order (KKT) conditions of all agents can be assembled into a Mixed Complementarity Problem (MCP). When all feasible sets are polyhedral and cost functions are convex, the MOPEC is amenable to product-space quasi-variational inequality (QVI) reformulations.

2. Illustrative Models and Application Domains

In networked mobility, Bandiera et al. formulate a model with $K$ competing Mobility Service Providers (MSPs), each optimizing a profit functional $\Pi^k(\mathbf{x})$ over strategic variables $x^k$, where user route choices form a lower-level equilibrium (network VI) that depends non-separably on the MSPs' decisions and the induced link flows. The overall problem is: $$\text{for }k=1,...,K: \quad \max_{x^k\in X^k} \Pi^k(\mathbf{x}) \quad \text{s.t. } f^*\in \arg\mathrm{VI}[F(\mathbf{x}), c(\cdot;\mathbf{x})]$$ with users' equilibrium flows $f^*$ encoding the Wardrop principle under potentially non-additive generalized costs (Bandiera et al., 2023).

In economic equilibrium, a standard MOPEC consists of $N$ producers (optimization agents) and market-clearing conditions (VI agent), as in the Mathiesen model implemented in GAMS/EMP, encompassing utility maximization, production under no-arbitrage, and market price equilibrium as a system (Kim et al., 2018).

Polynomial GNEPs with block-linear constraints are also MOPECs: each player's minimization of a polynomial objective over constraints linear in their strategy, with global Nash equilibrium constraints on the overall feasible set, admitting KKT-based branch decomposition and solution via Moment-SOS relaxations (Choi et al., 2024).

3. Solution Algorithms and Computational Approaches

MOPECs pose significant computational challenges due to nonconvexity, nonseparability, and equilibrium-constraint complexity. State-of-the-art approaches include:

• Nested Decomposition/Projection Methods: An outer SQP (Sequential Quadratic Programming) or first-order update on the upper-level agents' controls, coupled with repeated solution of the lower-level VI/equilibrium using projection or extragradient algorithms (Modified Projection Method/Extragradient) (Bandiera et al., 2023).
• Extended Mathematical Programming (EMP): EMP in GAMS provides high-level constructs for MOPEC, GNEP, and VI modeling, with automated construction of MCP or QVI representations, support for shared variables, and several sparse/implicit reformulations to control model size and trade-off between redundancy and sparsity (Kim et al., 2018).
• Hierarchy-Free Approximations: For bilevel or multi-leader MOPECs, T-step Cournot (best-response dynamics) and monopoly (jointly optimized) models yield upper and lower bounds to the true bilevel value, converging exponentially in T; these can be solved efficiently with first-order methods and avoid hierarchical AD (Li et al., 2023).
• Moment-SOS Hierarchy for Polynomial Branches: Polynomial GNEPs with quasi-linear constraints are decomposed into branches via Carathéodory's theorem; each branch polynomial program is solved globally by Lasserre's Moment-SOS relaxations, ensuring completeness and global optimality for all generalized Nash equilibria (Choi et al., 2024).
• Distributed/Zeroth-Order Gradient Tracking: For large-scale, networked, or stochastic MOPECs, distributed implicit gradient/zeroth-order methods using random smoothing, finite-difference, and inexact lower-level VI solutions achieve optimal rates for nonsmooth stochastic nonconvex problems both in theory and in numerical performance (Ebrahimi et al., 28 May 2025, Cui et al., 2021).

4. Theoretical Foundations: Optimality and Error-Bound Analysis

Recent advances have extended first-order and stationarity theory for MOPECs, even under nonsmooth and vector equilibrium constraints. Necessary optimality conditions can be derived via penalization and error-bound arguments. If a residual $\sigma(\xi, x)$ quantifies violation of the equilibrium condition, and an error bound establishes $$\mathrm{dist}((\xi, x), \text{gph}~SE)\leq \gamma \sigma(\xi, x)$$ near a solution, then penalty reformulations deliver exact minimization for sufficiently large penalty parameter (Uderzo, 2022).

Under subdifferential/normal qualification and metric regularity, first-order conditions take the form: $$0\in \partial \phi(\bar \xi, \bar x) + \lambda \left[(N_Q(\bar \xi)\cap B)\times\{0\}\right] + \lambda \left[\partial_x \nu(\bar \xi, \bar x) + D^*K(\bar \xi, \bar x)(B)\times B\right]$$ for $\lambda>\gamma\cdot \mathrm{Lip}(\phi)$, relating the subgradient, normal cones, and coderivatives of the feasible mappings (Uderzo, 2022).

Error bound and regularity conditions (via subtransversality or metric regularity) enable practical penalty and augmented Lagrangian methods, with performance guarantees.

5. Modeling Frameworks and Unified Solution Concepts

MOPECs admit unification under frameworks such as:

• Mathematical Program Networks (MPNs): Laine's MPN abstraction treats each agent/problem as a node in a directed-acyclic dependency network, with solution graphs encoding hierarchical or cooperative constraints. The equilibrium of an MPN is the intersection of all nodes' local solution graphs, and this viewpoint recovers MOPECs, GNEPs, Stackelberg games, and EPECs as special acyclic MPNs (Laine, 2024).
• EMP/GAMS: The EMP framework allows MOPECs and GNEP-like variational problems to be declared in near-algebraic syntax, with "equilibrium," "max"/"min," "vi," and "implicit" tags guiding the reformulation to MCP or QVI systems, facilitating solution using complementarity solvers or MIP methods (Kim et al., 2018).
• Product-Space QVI and MCP: Many MOPECs, under convexity and polyhedrality, are reducible to large-scale MCPs, for which specialized solvers and modeling languages automate derivative computation and sparse handling of shared variables (Kim et al., 2018, Choi et al., 2024).

6. Numerical Results and Practical Impact

Empirical investigations have demonstrated both the power and complexity of MOPEC methods:

• In multi-modal transportation equilibrium with multiple MSPs, non-separability of link costs produces up to 5% profit errors when neglected, joint subscription products can increase profits by 20%, and competition dynamically alters equilibrium fleet sizes (Bandiera et al., 2023).
• In distributed SMPECs, network topology affects convergence speed, with denser graphs achieving consensus and objective convergence more rapidly. Distributed implicit zeroth-order methods achieve centralized performance without full information aggregation, matching or improving sample/policy complexity (Ebrahimi et al., 28 May 2025).
• Variance-reduced stochastic proximal-point schemes (VR-SPP) and smoothed proximal best-response (SPBR) can attain two orders of magnitude accuracy improvements and up to 20× faster computation on multi-leader multi-follower hierarchical games compared to classical stochastic approximation (Cui et al., 2021).
• Polynomial GNEPs with up to $\sim 20$ constraints and hundreds of branches are solvable in minutes via the branch- and-Moment-SOS pipeline, providing full enumeration of isolated Nash equilibria or certificates of nonexistence (Choi et al., 2024).

7. Limitations, Open Problems, and Future Directions

MOPECs are computationally demanding due to combinatorial explosion of branch enumeration (e.g., Carathéodory decomposition), nonconvex coupling in upper-level games, and large-scale variational subproblems. Solution existence, uniqueness, and global optimality are generally not guaranteed beyond convexity and monotonicity regimes.

Open challenges include extending efficient algorithms to cyclic and feedback networks, handling nonconvex and nonmonotone lower-level (VI/complementarity) structures, exploiting inexact or partial-information solvers in settings with uncertainty, and developing robust, scalable decomposition methods for high-dimensional and dynamic MOPECs.

Promising directions are advanced decomposition schemes (e.g., progressive leader-follower or block-wise Gauss-Seidel), inexact VI/bundle methods, and single-level reformulations via complementarity or D-gap functions (enabling global MINLP solvers), as well as further exploiting the MPN abstraction for structure-exploiting solution algorithms in high-dimensional or distributed multi-agent systems (Bandiera et al., 2023, Laine, 2024).