Newton-Type Methods for Optimization Problems without Constraint Qualifications

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.

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

We establish local convergence and rate of convergence of the classical augmented Lagrangian algorithm under the sole assumption that the dual starting point is close to a multiplier satisfying the second-order sufficient optimality condition. In particular, no constraint qualifications of any kind are needed. Previous literature on the subject required, in addition, the linear independence constraint qualification and either the strict complementarity assumption or a stronger version of the second-order sufficient condition. That said, the classical results allow the initial multiplier estimate to be far from the optimal one, at the expense of proportionally increasing the threshold value for the penalty parameters. Although our primary goal is to avoid constraint qualifications, if the stronger assumptions are introduced, then starting points far from the optimal multiplier are allowed within our analysis as well. Using only the second-order sufficient optimality condition, for penalty parameters large enough we prove primal-dual Q-linear convergence rate, which becomes superlinear if the parameters are allowed to go to infinity. Both exact and inexact solutions of subproblems are considered. In the exact case, we further show that the primal convergence rate is of the same Q-order as the primal-dual rate. Previous assertions for the primal sequence all had to do with the weaker R-rate of convergence and required the stronger assumptions cited above. Finally, we show that under our assumptions one of the popular rules of controlling the penalty parameters ensures their boundedness.

The paper presents a new approach to solving nonlinear programming (NLP) problems for which the strict complementarity condition (SCC), a constraint qualification (CQ), and a second-order sufficient condition (SOSC) for optimality are not necessarily satisfied at a solution. Our approach is based on the construction of p-regularity and on reformulating the inequality constraints as equalities. Namely, by introducing the slack variables, we get the equality constrained problem, for which the Lagrange optimality system is singular at the solution of the NLP problem in the case of the violation of the CQs, SCC and/or SOSC. To overcome the difficulty of singularity, we propose the p-factor method for solving the Lagrange system. The method has a superlinear rate of convergence under a mild assumption. We show that our assumption is always satisfied under a standard second-order sufficient condition (SOSC) for optimality. At the same time, we give examples of the problems where the SOSC does not hold, but our assumption is satisfied. Moreover, no estimation of the set of active constraints is required. The proposed approach can be applied to a variety of problems.

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.

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...

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.

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.

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 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.

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.

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.

We consider a class of mathematical programs with complementarity constraints (MPCC) where the objective function involves a non-Lipschitz sparsity-inducing term. Due to the existence of the non-Lipschitz term, existing constraint qualifications for locally Lipschitz MPCC cannot ensure that necessary optimality conditions hold at a local minimizer. In this paper, we present necessary optimality conditions and MPCC-tailored qualifications for the non-Lipschitz MPCC. The proposed qualifications are related to the constraints and the non-Lipschitz term, which ensure that local minimizers satisfy these necessary optimality conditions. Moreover, we present an approximation method for solving the non-Lipschitz MPCC and establish its convergence. Finally, we use numerical examples of sparse solutions of linear complementarity problems and the second-best road pricing problem in transportation science to illustrate the effectiveness of our approximation method for solving the non-Lipschitz MPCC.

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.

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.