Implementing Maximum Likelihood Estimation of Empirical Matching Models: MPEC v.s. NFXP

This book provides a solid foundation and an extensive study for an important class of constrained optimization problems known as Mathematical Programs with Equilibrium Constraints (MPEC), which are extensions of bilevel optimization problems. The book begins with the description of many source problems arising from engineering and economics that are amenable to treatment by the MPEC methodology. Error bounds and parametric analysis are the main tools to establish a theory of exact penalisation, a set of MPEC constraint qualifications and the first-order and second-order optimality conditions. The book also describes several iterative algorithms such as a penalty-based interior point algorithm, an implicit programming algorithm and a piecewise sequential quadratic programming algorithm for MPECs. Results in the book are expected to have significant impacts in such disciplines as engineering design, economics and game equilibria, and transportation planning, within all of which MPEC has a central role to play in the modelling of many practical problems.

This paper concentrates on improving the convergence properties of the relaxation schemes introduced by Kadrani et al. and Kanzow and Schwartz for mathematical program with equilibrium constraints (MPEC) by weakening the original constraint qualifications. It has been known that MPEC relaxed constant positive-linear dependence (MPEC-RCPLD) is a class of extremely weak constraint qualifications for MPEC, which can be strictly implied by MPEC relaxed constant rank constraint qualification (MPEC-RCRCQ) and MPEC relaxed constant positive-linear dependence (MPEC-rCPLD), of course also by the MPEC constant positive-linear dependence (MPEC-CPLD). We show that any accumulation point of stationary points of these two approximation problems is M-stationarity under the MPEC-RCPLD constraint qualification, and further show that the accumulation point can even be S-stationarity coupled with the asymptotically weak nondegeneracy condition.

Mathematical programs with equilibrium constraints (MPECs) represent an important class of nonlinear optimization problems. Due to their constraint set being defined as the solution of some parameter-dependent generalized equation, the application of standard constraint qualifications (CQs) from nonlinear programming to MPECs is not straightforward. Rather than turning MPECs into mathematical programs with complementarity constraints (MPCCs) and applying specially adapted CQs, we want to present here a variational-analytic approach to dual stationarity conditions for MPECs on the basis of Lipschitzian properties of the perturbed generalized equation. The focus will be on the so-called calmness property, ensuring an appropriate calculus rule for the Mordukhovich normal cone.

In this chapter we are concerned with the problem known as a mathematical program with equilibrium constraints (MPEC) and with its game theoretic counterpart – the Stackelberg game . MPECs and Stackelberg games arise in many branches of engineering, science, and economics. MPEC network applications include certain network design , plant location , congestion pricing, and tolling problems formulated so that a fixed-point, nonlinear complementarity, variational inequality, or other mathematical representation of an equilibrium acts as a set of constraints on the optimizing intentions of a central authority. In particular, an MPEC is based on a partition of decision variables into two sets: those variables that are directly determined by a “leader” or omniscient agent, and those variables that are determined by a noncooperative Nash game among so-called “followers” who take the leader’s strategy as given. That is, the leader’s actions are beyond the followers’ influence.

Given a suitably parametrized family of equilibrium models and a higher level criterion by which to measure an equilibrium state, mathematical programs with equilibrium constraints (MPECs) provide a framework for improving or optimizing the equilibrium state. An example is toll design in traffic networks, which attempts to reduce total travel time by choosing which arcs to toll and what toll levels to impose. Here, a Wardrop equilibrium describes the traffic response to each toll design. Communication networks also have a deep literature on equilibrium flows that suggest some MPECs. We focus on mathematical programs with complementarity constraints (MPCCs), a subclass of MPECs for which the lower level equilibrium system can be formulated as a complementarity problem and therefore, importantly, as a nonlinear program (NLP). Although MPECs and MPCCs are typically non-convex, which is a consequence of the upper level objective clashing with the users' objectives in the lower level equilibrium program, the last decade of research has paved the way for finding local solutions of MPCCs via standard NLP techniques.

Generalized stationary points of the mathematical program with equilibrium constraints (MPEC) are studied to better describe the limit points produced by interior point methods for MPEC. A primal-dual interior-point method is then proposed, which solves a sequence of relaxed barrier problems derived from MPEC. Global convergence results are deduced under fairly general conditions other than strict complementarity or the linear independence constraint qualification for MPEC (MPEC-LICQ). It is shown that every limit point of the generated sequence is a strong stationary point of MPEC if the penalty parameter of the merit function is bounded. Otherwise, a point with certain stationarity can be obtained. Preliminary numerical results are reported, which include a case analyzed by Leyffer for which the penalty interior-point algorithm failed to find a stationary point.

Mathematical programs with equilibrium constraints (MPEC) are nonlinear programs which do not satisfy any of the common constraint qualifications (CQ). In order to obtain first-order optimality conditions, constraint qualifications tailored to the MPECs have been developed and researched in the past. In this paper, we introduce a new Abadie-type constraint qualification for MPECs. We investigate sufficient conditions for this new CQ, discuss its relationship to several existing MPEC constraint qualifications, and introduce a new Slater-type constraint qualifications. Finally, we prove a new stationarity concept to be a necessary optimality condition under our new Abadie-type CQ.

This chapter documents how to formulate mathematical programs with equilibrium constraints (MPECs), which naturally arise in a wide range of engineering and economic systems. We describe Pyomo components for complementarity conditions, and transformation capabilities that automate the reformulation of MPEC models, and meta-solvers that leverage these transformations to support global and local optimization of MPEC models.

We study theoretical and computational aspects of an exact penalization approach to mathematical programs with equilibrium constraints (MPECs). In the first part, we prove that a Mangasarian--Fromovitz-type condition ensures the existence of a stable local error bound at the root of a real-valued nonnegative piecewise smooth function. A specification to nonsmooth formulations of equilibrium constraints, e.g., complementarity conditions or normal equations, provides conditions which guarantee the existence of a nonsmooth exact penalty function for MPECs. In the second part, we study a trust region minimization method for a class of composite nonsmooth functions which comprises exact penalty functions arising from MPECs. We prove a global convergence result for the general method and incorporate a penalty update rule. A further specification results in an SQP trust region method for MPECs based on an \(\ell_1\) penalty function.

An MPEC formulation for dynamic optimization of distillation operations

Information security is a critical yet challenging issue for wireless communications. In this paper, we consider the distributed resource competition in a network that consists of both security-oriented users (SeUs) and regular users (ReUs) which, respectively, intend for secrecy rate and transmission rate maximization. To enhance wireless security, the SeUs are given higher priorities such that they are allowed to take action first in the competition, which gives rise to the multi-leader-follower hierarchical game formulation where the SeUs are the leaders in the upper layer and ReUs are the followers in the lower layer. However, the solution to the lower sub-game among the ReUs, in the form of a Nash equilibrium parameterized by the upper strategy, lacks closed-form expression, which hinders us from solving the hierarchical game effectively. To tackle this issue, we first consider the case with one leader and reformulate the game as a mathematical program with equilibrium constraints (MPEC). Then, the MPEC is transformed as a single-level optimization and solved through successive concave approximation. For the general case that comprises multiple leaders, the equilibrium program with equilibrium constraints (EPEC) is introduced for the game reformulation. Due to the inherent difficulties of EPEC, the relaxed concept of local Nash equilibrium (LNE) is introduced as the solution. Furthermore, the existence and uniqueness of the LNE are investigated with the variational inequality-based analysis. Finally, simulation results are provided to corroborate our theoretical findings.

Recently, nonlinear programming solvers have been used to solve a range of mathematical programs with equilibrium constraints (MPECs). In particular, sequential quadratic programming (SQP) methods have been very successful. This paper examines the local convergence properties of SQP methods applied to MPECs. SQP is shown to converge superlinearly under reasonable assumptions near a strongly stationary point. A number of examples are presented that show that some of the assumptions are difficult to relax.

Stackelberg games and multiple equilibrium behaviors on networks

This paper presents a mathematical programming based technique for the minimum weight (volume) topology optimization of truss-like structures such that strain softening material properties, that can lead to severe physical instability, can be accommodated. In addition, satisfaction of such serviceability criteria as limited displacements at some specific points is ensured. The problem is formulated in terms of truss member cross-sectional areas. This leads to a challenging nonconvex and nonsmooth optimization problem, known as a mathematical program with equilibrium constraints (MPEC). A two-step optimization algorithm is proposed to overcome the problems typically associated with nondefiniteness of some key matrices and at the same time nondifferentiability of the mathematical system. Each step involves updating the ground structure and solving the MPEC using a penalized nonlinear programming (NLP) approach. Some numerical examples are provided to illustrate application, robustness and efficiency of the proposed scheme. The safety and integrity of the designed topologically optimal structures are validated using appropriate stepwise holonomic elastoplastic analyses.

We revisit an improtant and challenging class of minimum weight structural plasticity problems, the feature of which is the presence of complementarity constraints. Such relations mathematically express the perpendicularity of two sign-constrained vectors and mechanically describe an inherent property of plasticity. The optimization problem in point is referred to in the mathematical programming literature as a Mathematical Program with Equilibrium Constraints (MPEC). Due to its intrinsic complexity, MPECs are computationally very hard to solve. In this paper, we adopt recent ideas, proposed by mathematical programmers, on smoothing to develop a simple scheme for reformulating and solving our minimum weight problem as a standard nonlinear program. Simple examples concerning truss-like structures are also presented for illustrative purposes.