Mathematical programs with complementarity constraints in traffic and telecommunications networks

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.

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.

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.

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.

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.

An MPEC formulation for dynamic optimization of distillation operations

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.

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

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.

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.

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.

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.

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.

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.

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.