Complementarity Constraints Research Articles

On linear problems with complementarity constraints

A mathematical program with complementarity constraints is an optimization problem with equality/inequality constraints in which a complementarity type constraint is considered in addition. This complementarity condition modifies the feasible region so as to remove many of those properties that are usually important to obtain the standard optimality conditions, e.g., convexity and constraint qualifications. In this paper, in the linear case, we introduce a decomposition method of the given problem in a sequence of parameterized problems, that aim to force complementarity. Once we obtain a feasible solution, by means of duality results, we are able to eliminate a set of parameterized problems which are not worthwhile to be considered. Furthermore, we provide some bounds for the optimal value of the objective function and we present an application of the proposed technique in a non trivial example and some numerical experiments.

Sequential Linearization Method for Bound-Constrained Mathematical Programs with Complementarity Constraints

Sequential Linearization Method for Bound-Constrained Mathematical Programs with Complementarity Constraints

Mediating Between Contact Feasibility and Robustness of Trajectory Optimization Through Chance Complementarity Constraints.

As robots move from the laboratory into the real world, motion planning will need to account for model uncertainty and risk. For robot motions involving intermittent contact, planning for uncertainty in contact is especially important, as failure to successfully make and maintain contact can be catastrophic. Here, we model uncertainty in terrain geometry and friction characteristics, and combine a risk-sensitive objective with chance constraints to provide a trade-off between robustness to uncertainty and constraint satisfaction with an arbitrarily high feasibility guarantee. We evaluate our approach in two simple examples: a push-block system for benchmarking and a single-legged hopper. We demonstrate that chance constraints alone produce trajectories similar to those produced using strict complementarity constraints; however, when equipped with a robust objective, we show the chance constraints can mediate a trade-off between robustness to uncertainty and strict constraint satisfaction. Thus, our study may represent an important step towards reasoning about contact uncertainty in motion planning.

A Sequential Convex Programming Approach to Solving Quadratic Programs and Optimal Control Problems With Linear Complementarity Constraints

Mathematical programs with complementarity constraints are notoriously difficult to solve due to their nonconvexity and lack of constraint qualifications in every feasible point. This letter focuses on the subclass of quadratic programs with linear complementarity constraints. A novel approach to solving a penalty reformulation using sequential convex programming and a homotopy on the penalty parameter is introduced. Linearizing the necessarily nonconvex penalty function yields convex quadratic subproblems, which have a constant Hessian matrix throughout all iterates. This allows solution computation with a single KKT matrix factorization. Furthermore, a globalization scheme is introduced in which the underlying merit function is minimized analytically, and guarantee of descent is provided at each iterate. The algorithmic features and possible computational speedups are illustrated in a numerical experiment.

On the Karush-Kuhn-Tucker reformulation of the bilevel optimization problems on Riemannian manifolds

Bilevel programming problems are often reformulated using the Karush-Kuhn-Tucker conditions for the lower level problem resulting in a mathematical program with complementarity constraints (MPCC). First, we present KKT reformulation of the bilevel optimization problems on Riemannian manifolds. Moreover, we show that global optimal solutions of theMPCCcorrespond to global optimal solutions of the bilevel problem on the Riemannian manifolds provided the lower level convex problem satisfies the Slater?s constraint qualification. But the relationship between the local solutions of the bilevel problem and its corresponding MPCC is incomplete equivalent. We then also show by examples that these correspondences can fail if the Slater?s constraint qualification fails to hold at lower-level convex problem. In addition,M- and C-type optimality conditions for the bilevel problem on Riemannian manifolds are given.

On linear bilevel optimization problems with complementarity-constrained lower levels

We consider a novel class of linear bilevel optimization models with a lower level that is a linear program with complementarity constraints (LPCC). We present different single-level reformulations depending on whether the linear complementarity problem (LCP) as part of the lower-level constraint set depends on the upper-level decisions or not as well as on whether the LCP matrix is positive definite or positive semidefinite. Moreover, we illustrate the connection to linear trilevel models that can be reduced to bilevel problems with LPCC lower levels having positive (semi)definite matrices. Finally, we provide two generic and illustrative bilevel models from the fields of transportation and energy to show the practical relevance of the newly introduced class of bilevel problems and show related theoretical results.

Gas Network's Impact on Power System Voltage Security

Due to the energy linkage between electricity and gas networks, assessing the voltage security of the electricity system without considering the practical constraints of both systems, may lead to unrealistic values of loading margins (LM). This work proposes a model for investigating the impact of gas networks on the voltage security of electric transmission networks. The overall objective is to maximize the LM of the electricity network while satisfying all relevant constraints in both gas and electricity networks such as hourly line pack of gas pipelines, reactive power capability limits of generators, and complementarity constraints representing the generators active/reactive power limits based on the capability curves, power flow equations at both current operation, and security limit points. Three (small, medium, and large) case studies are presented as the applications of the proposed model for LM maximization in power systems that are highly coupled with gas networks. The obtained results corroborate the impact of both gas and electrical networks operation constraints such as voltage and reactive power limits, nodal gas pressure limits, gas network loading as well as the line pack phenomenon of gas pipelines on the LM of power systems.

A new elementary proof for M-stationarity under MPCC-GCQ for mathematical programs with complementarity constraints

It is known in the literature that local minimizers of mathematical programs with complementarity constraints (MPCCs) are so-called M-stationary points, if a weak MPCC-tailored Guignard constraint qualification (called MPCC-GCQ) holds. In this paper we present a new elementary proof for this result. Our proof is significantly simpler than existing proofs and does not rely on deeper technical theory such as calculus rules for limiting normal cones. A crucial ingredient is a proof of a (to the best of our knowledge previously open) conjecture, which was formulated in a Diploma thesis by Schinabeck.

Pure characteristics demand models and distributionally robust mathematical programs with stochastic complementarity constraints

We formulate pure characteristics demand models under uncertainties of probability distributions as distributionally robust mathematical programs with stochastic complementarity constraints (DRMP-SCC). For any fixed first-stage variable and a random realization, the second-stage problem of DRMP-SCC is a monotone linear complementarity problem (LCP). To deal with ambiguity of probability distributions of the involved random variables in the stochastic LCP, we use the distributionally robust approach. Moreover, we propose an approximation problem with regularization and discretization to solve DRMP-SCC, which is a two-stage nonconvex-nonconcave minimax optimization problem. We prove the convergence of the approximation problem to DRMP-SCC regarding the optimal solution sets, optimal values and stationary points as the regularization parameter goes to zero and the sample size goes to infinity. Finally, preliminary numerical results for investigating distributional robustness of pure characteristics demand models are reported to illustrate the effectiveness and efficiency of our approaches.

A New Augmented Lagrangian Method for MPCCs—Theoretical and Numerical Comparison with Existing Augmented Lagrangian Methods

We propose a new augmented Lagrangian (AL) method for solving the mathematical program with complementarity constraints (MPCC), where the complementarity constraints are left out of the AL function and treated directly. Two observations motivate us to propose this method: The AL subproblems are closer to the original problem in terms of the constraint structure; and the AL subproblems can be solved efficiently by a nonmonotone projected gradient method, in which we have closed-form solutions at each iteration. The former property helps us show that the proposed method converges globally to an M-stationary (better than C-stationary) point under MPCC relaxed constant positive linear dependence condition. Theoretical comparison with existing AL methods demonstrates that the proposed method is superior in terms of the quality of accumulation points and the strength of assumptions. Numerical comparison, based on problems in MacMPEC, validates the theoretical results.

A new mathematical program with complementarity constraints for optimal localization of pressure reducing valves in water distribution systems

Water loss reduction in water distribution systems (WDSs) is a challenging task for water utilities worldwide. One of the most reliable and cost-effective ways to reduce water loss is to properly regulate operational pressure of the system through optimizing pressure reducing valve (PRV) placements. This well-known engineering problem can be casted into a mixed-integer nonlinear program (MINLP) where binary variables are introduced to represent positions of PRVs. Many works in the literature applied heuristic algorithms to address the optimization problem. In this paper, at first, we proposed a new optimization model and reformulated it as the mathematical program with complementarity constraints (MPCCs). It is due to the fact that the stationary point of the MPCCs is likely to be trapped into bad local solutions, a soft heuristic method is then proposed to determine the MINLP local solution in each iteration before a stationary point of the MPCCs is reached. This method not only enhances the quality of MINLP solution, but also decreases computation time for solving the MPCCs. The newly formulated MPCCs is applied to determine optimal localization of PRVs for two WDS benchmarks and a real-world WDS in Vietnam. The results are compared with others in the literature demonstrating that using our new optimization model, better and more reliable MINLP solution can be found for large scale WDSs.

Robust calculation method of voltage stability margin based on voltage‐collapse‐point properties

The progress in electricity deregulation has made electricity systems more economical while maintaining stability. Therefore, the stability of an electricity system should be evaluated more strictly than done previously. We propose a robust calculation method of voltage stability margin (VSM). The proposed method is based on the optimal‐power‐flow‐based direct method (OPF‐DM), which can calculate VSM directly when complementarity constraints related to limits for reactive‐power generation are added. However, the additional complementarity constraints may cause a convergence and local solution issues because of the numerical complexity. The proposed method calculates the appropriate VSM on the basis of several voltage‐stability properties. We implemented the proposed method on MATPOWER and evaluated it with IEEE 9‐bus, 30‐bus system, 39‐bus, 300‐bus, PEGASE 1354‐bus, and IEEJ WEST10‐O/V systems. The results indicate that the proposed method can calculate VSM robustly compared with conventional methods. © 2021 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

Sequential optimality conditions for cardinality-constrained optimization problems with applications

Recently, a new approach to tackle cardinality-constrained optimization problems based on a continuous reformulation of the problem was proposed. Following this approach, we derive a problem-tailored sequential optimality condition, which is satisfied at every local minimizer without requiring any constraint qualification. We relate this condition to an existing M-type stationary concept by introducing a weak sequential constraint qualification based on a cone-continuity property. Finally, we present two algorithmic applications: We improve existing results for a known regularization method by proving that it generates limit points satisfying the aforementioned optimality conditions even if the subproblems are only solved inexactly. And we show that, under a suitable Kurdyka–Łojasiewicz-type assumption, any limit point of a standard (safeguarded) multiplier penalty method applied directly to the reformulated problem also satisfies the optimality condition. These results are stronger than corresponding ones known for the related class of mathematical programs with complementarity constraints.

Stochastic mathematical programs with probabilistic complementarity constraints: SAA and distributionally robust approaches

Stochastic mathematical programs with probabilistic complementarity constraints: SAA and distributionally robust approaches

Second-order Optimality Conditions for Mathematical Program with Semidefinite Cone Complementarity Constraints and Applications

Second-order Optimality Conditions for Mathematical Program with Semidefinite Cone Complementarity Constraints and Applications

MPCC: Strong Stability of M-stationary Points

In this paper we study the class of mathematical programs with complementarity constraints MPCC. Under the Linear Independence constraint qualification MPCC-LICQ we state a topological as well as an equivalent algebraic characterization for the strong stability (in the sense of Kojima) of an M-stationary point for MPCC. By allowing perturbations of the describing functions up to second order, the concept of strong stability refers here to the local existence and uniqueness of an M-stationary point for any sufficiently small perturbed problem where this unique solution depends continuously on the perturbation. Finally, some relations to S- and C-stationarity are briefly discussed.

Strategic Interactions between Liquefied Natural Gas and Domestic Gas Markets: A Bilevel Model

This paper presents a bilevel programming model to aid decision-making for two players who interact strategically in the liquefied natural gas (LNG) supply chain, possibly with conflicting interests. In the proposed model, an LNG operator is the leader and a natural gas (NG) producer is the follower. The LNG operator attempts to optimally locate LNG export terminals, purchase gas from the NG producer, and export it as LNG. The NG producer aims to optimize production, pipeline investments, and sales to domestic consumers and the LNG operator. To solve the bilevel problem, we first reformulate it as a single-level problem by exploiting the convexity of the lower-level problem. Then, we use disjunctive reformulations of complementarity constraints and piecewise linear approximations of objective function terms to convert the problem into a convex quadratic mixed-integer program (QMIP). Computational experiments confirm that the QMIP is tractable and can be solved efficiently. We apply our bilevel framework to a case study of the Gulf-Southwest region of the United States and evaluate several decision-making scenarios. The scenario results emphasize the importance of using the bilevel methodology to anticipate the effects of new LNG export facilities on NG prices across the domestic gas network. Adding these large gas-consuming facilities at specific locations puts upward pressure on domestic gas prices, although this is somewhat mitigated by the NG producer's optimal production response to the increased demand.

Local Stackelberg Equilibrium Seeking in Generalized Aggregative Games

We propose a two-layer, semi-decentralized algorithm to compute a local solution to the Stackelberg equilibrium problem in aggregative games with coupling constraints. Specifically, we focus on a single-leader, multiple-follower problem, and after equivalently recasting the Stackelberg game as a mathematical program with complementarity constraints (MPCC), we iteratively convexify a regularized version of the MPCC as inner problem, whose solution generates a sequence of feasible descent directions for the original MPCC. Thus, by pursuing a descent direction at every outer iteration, we establish convergence to a local Stackelberg equilibrium. Finally, the proposed algorithm is tested on a numerical case study involving a hierarchical instance of the charging coordination of Plug-in Electric Vehicles (PEVs).

Global solutions of nonconvex standard quadratic programs via mixed integer linear programming reformulations

A standard quadratic program is an optimization problem that consists of minimizing a (nonconvex) quadratic form over the unit simplex. We focus on reformulating a standard quadratic program as a mixed integer linear programming problem. We propose two alternative formulations. Our first formulation is based on casting a standard quadratic program as a linear program with complementarity constraints. We then employ binary variables to linearize the complementarity constraints. For the second formulation, we first derive an overestimating function of the objective function and establish its tightness at any global minimizer. We then linearize the overestimating function using binary variables and obtain our second formulation. For both formulations, we propose a set of valid inequalities. Our extensive computational results illustrate that the proposed mixed integer linear programming reformulations significantly outperform other global solution approaches. On larger instances, we usually observe improvements of several orders of magnitude.

A unified framework for high-order numerical discretizations of variational inequalities

We present in this work a unified framework for elliptic variational inequalities that gathers several problems in contact mechanics like the unilateral contact of one or two membranes or the Signorini problem. We study a family of Galerkin numerical schemes that discretize this framework. We prove the well-posedness of the discrete problem and we show that it is equivalent to a saddle-point mixed formulation containing complementarity constraints. To solve the arising nonlinear problem, we employ a semismooth Newton method and we prove local convergence properties. The abstract framework is then applied to the discretization of the unilateral contact between two membranes. We propose to discretize this problem with a finite element (FEM), a discontinuous Galerkin (dG), and a hybrid high-order (HHO) methods. We also adapt the semismooth Newton algorithm, including a static condensation procedure for the HHO method. Finally, we run numerical experiments for the FEM and HHO discretizations and compare their behavior.