Mathematical programs with complementarity constraints and a non-Lipschitz objective: optimality and approximation

In this article, a smooth mathematical program with complementarity constraints (MPCC) is reformulated as a non-smooth constrained optimization problem by using the Fischer-Burmeister function. A lower-order penalty method is applied to transform the resulted constrained optimization problem into unconstrained optimization problems. Lower-order penalty functions may not be locally Lipschitz. However, they require weaker conditions to guarantee an exact penalization property than the classical l 1 penalty functions. We derive optimality conditions for the penalty problems using a smooth approximate variational principle, and establish that the limit point of a sequence of points that satisfy the second-order necessary optimality conditions of penalty problems is a strongly stationary point (hence a B-stationary point) of the original MPCC if the limit point is feasible to MPCC, and a linear independence constraint qualification for MPCC and an upper level strict complementarity condition hold at the limit point. Furthermore, the limit point also satisfies a second-order necessary condition of MPCC. Numerical examples are presented to demonstrate and compare the effectiveness of the proposed methods.

Mathematical programs with complementarity constraints (MPCC) is an important subclass of MPEC, and for conventional MPEC, we can transform it into the MPCC form in some manner. It is a nature way to solve MPCC by constructing a suitable approximation of the primal problem. In this paper, we present a class of smoothing methods for MPCC, it is a broader approximation, and by selecting an available probability density function, we can obtain a corresponding approximation of MPCC. We show that the linear independence constraint qualification holds for the class of smooth methods under some conditions. We also analyze the convergence properties of the accumulated point gotten by the class of smooth methods.

Continuous network design problem (CNDP) is searching for a transportation network configuration to minimize the sum of the total system travel time and the investment cost of link capacity expansions by considering that the travellers follow a traditional Wardrop user equilibrium (UE) to choose their routes. In this paper, the CNDP model can be formulated as mathematical programs with complementarity constraints (MPCC) by describing UE as a non-linear complementarity problem (NCP). To address the difficulty resulting from complementarity constraints in MPCC, they are substituted by the Fischer-Burmeister (FB) function, which can be smoothed by the introduction of the smoothing parameter. Therefore, the MPCC can be transformed into a well-behaved non-linear program (NLP) by replacing the complementarity constraints with a smooth equation. Consequently, the solver such as LINDOGLOBAL in GAMS can be used to solve the smooth approximate NLP to obtain the solution to MPCC for modelling CNDP. The numerical experiments on the example from the literature demonstrate that the proposed algorithm is feasible.

To reflect uncertain data in practical problems, stochastic versions of the mathematical program with complementarity constraints (MPCC) have drawn much attention in the recent literature. Our concern is the detailed analysis of convergence properties of a regularization sample average approximation (SAA) method for solving a stochastic mathematical program with complementarity constraints (SMPCC). The analysis of this regularization method is carried out in three steps: First, the almost sure convergence of optimal solutions of the regularized SAA problem to that of the true problem is established by the notion of epiconvergence in variational analysis. Second, under MPCC-MFCQ, which is weaker than MPCC-LICQ, we show that any accumulation point of Karash-Kuhn-Tucker points of the regularized SAA problem is almost surely a kind of stationary point of SMPCC as the sample size tends to infinity. Finally, some numerical results are reported to show the efficiency of the method proposed.

This paper examines solution methods for mathematical programs with complementarity constraints (MPCC) obtained from the time-discretization of optimal control problems (OCPs) subject to nonsmooth dynamical systems. The MPCC theory and stationarity concepts are reviewed and summarized. The focus is on relaxation-based methods for MPCCs, which solve a (finite) sequence of more regular nonlinear programs (NLP), where a regularization/homotopy parameter is driven to zero. Such methods perform reasonably well on currently available benchmarks. However, these results do not always generalize to MPCCs obtained from nonsmooth OCPs. To provide a more complete picture, this paper introduces a novel benchmark collection of such problems, which we call NOSBENCH. The problem set includes 603 different MPCCs and we split it into a few representative subsets to accelerate the testing. We compare different relaxation-based methods, NLP solvers, homotopy parameter update and relaxation parameter steering strategies. Moreover, we check whether the obtained stationary points allow first-order descent directions, which may be the case for some of the weaker MPCC stationarity concepts. In the best case, the Scholtes’ relaxation (SIAM J. Optim. 11, 918–936, 2001) with IPOPT (Math. Program. 106, 25–57, 2006) as NLP solver manages to solve 73.8% of the problems. This highlights the need for further improvements in algorithms and software for MPCCs.

In this paper, a mathematical program with complementarity constraints (MPCC) is reformulated as a nonsmooth constrained mathematical program via the Fischer–Burmeister function. Smooth penalty functions are used to treat this nonsmooth constrained program. Under linear independence constraint qualification, and upper level strict complementarity condition, together with some other mild conditions, we prove that the limit point of stationary points satisfying second-order necessary conditions of unconstrained penalized problems is a strongly stationary point, hence a B-stationary point of the original MPCC. Furthermore, this limit point also satisfies a second-order necessary condition of the original MPCC. Numerical results are presented to test the performance of this method.

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.

A new smoothing approach was given for solving the mathematical programs with complementarity constraints (MPCC) by using the aggregation technique. As the smoothing parameter tends to zero, if the KKT point sequence generated from the smoothed problems satisfies the second-order necessary condition, then any accumulation point of the sequence is a B-stationary point of MPCC if the linear independence constraint qualification (LICQ) and the upper level strict complementarity (ULSC) condition hold at the limit point. The ULSC condition is weaker than the lower level strict complementarity (LLSC) condition generally used in the literatures. Moreover, the method can be easily extended to the mathematical programs with general vertical complementarity constraints.

The exactness of the [formula omitted] penalty function for a class of mathematical programs with generalized complementarity constraints

We aim to compute lifted stationary points of a sparse optimization problem ( $$P_{0}$$ ) with complementarity constraints. We define a continuous relaxation problem ( $$R_{\nu }$$ ) that has the same global minimizers and optimal value with problem ( $$P_{0}$$ ). Problem ( $$R_{\nu }$$ ) is a mathematical program with complementarity constraints (MPCC) and a difference-of-convex objective function. We define MPCC lifted-stationarity of ( $$R_{\nu }$$ ) and show that it is weaker than directional stationarity, but stronger than Clarke stationarity for local optimality. Moreover, we propose an approximation method to solve ( $$R_{\nu }$$ ) and an augmented Lagrangian method to solve its subproblem ( $$R_{\nu ,\sigma }$$ ), which relaxes the equality constraint in ( $$R_{\nu }$$ ) with a tolerance $$\sigma $$ . We prove the convergence of our algorithm to an MPCC lifted-stationary point of problem ( $$R_{\nu }$$ ) and use a sparse optimization problem with vertical linear complementarity constraints to demonstrate the efficiency of our algorithm on finding sparse solutions in practice.

Combining the regularization strategy and the SQP to solve MPCC — A MATLAB implementation

In this article, we aim to find the most effective reformulation techniques to solve the MPCC (mathematical program with complementarity constraints) model that we proposed recently for continuous network design problems under asymmetric user equilibria. The MPCC model is based on a link-node nonlinear complementarity formulation for asymmetric user equilibria. By applying various reformulation techniques for the lower level nonlinear complementarity, the original bilevel formulation can be converted to a single level nonlinear programming problem. We show that certain reformulations are more effective than others to solve the proposed MPCC model. Recommendations are thus provided on how to choose a reformulation of the continuous network design problem that can be solved effectively and/or efficiently.

The generalized Nash equilibrium problem is an extension of the standard Nash equilibrium problem where both the utility function and the strategy space of each player depend on the strategies chosen by all other players. Recently, the generalized Nash equilibrium problem has emerged as an effective and powerful tool for modeling a wide class of problems arising in many fields and yet solution algorithms are extremely scarce. In this paper, using a regularized Nikaido-Isoda function, we reformulate the generalized Nash equilibrium problem as a mathematical program with complementarity constraints (MPCC). We then propose a suitable method for this MPCC and under some conditions, we establish the convergence of the proposed method by showing that any accumulation point of the generated sequence is a M-stationary point of the MPCC. Numerical results on some generalized Nash equilibrium problems are reported to illustrate the behavior of our approach.

Mathematical programs with complementarity constraints (MPCCs) are difficult optimization problems that do not satisfy the majority of the usual constraint qualifications (CQs) for standard nonline...

Using standard nonlinear programming (NLP) theory, we establish formulas for first and second order directional derivatives of optimal value functions of parametric mathematical programs with complementarity constraints (MPCCs). The main point is that under a linear independence condition on the active constraint gradients, optimal value sensitivity of MPCCs is essentially the same as for nonlinear programs, in spite of the combinatorial nature of the MPCC feasible set. Unlike NLP however, second order directional derivatives of the MPCC optimal value function show combinatorial structure.