Complementarity Conditions Research Articles

An exterior point algorithm for some linear complementarity problems with applications

An exterior point algorithm for positive definite (PD) and positive semidefinite (PSD) linear complementarity problems (LCPs) is introduced. The algorithm exploits the ellipsoid method to find a starting point in the case of positive definite linear complementarity problems (PDLCPs) and to check for the problem feasibility in case of positive semidefinite linear complementarity problems (PSDLCPs). The algorithm starts from a point on the boundary on which the complementarity condition is satisfied and generates a sequence of points on that same boundary. These points converge to the solution. The algorithm is modified to speed up the convergence for some PDLCPs and PSDLCPs that arise in certain mechanical models. A numerical example and a practical example in robotics are solved to test the algorithm.

Interpersonal Orientation in Context: Correlates and Effects of Interpersonal Complementarity on Subjective and Cardiovascular Experiences

Interpersonal orientation (IO) generally refers to individual differences in preference for social interaction. The influence of IO, however, likely depends on the nature of complementarity within the interpersonal context. Using the interpersonal circumplex and Five-Factor Model, we first selected a measure of IO characterized by affiliation and neuroticism. Second, we examined the influence of IO on subjective, physiological, and nonverbal experiences as a function of experimentally manipulated complementarity or noncomplementarity. We hypothesized that women in noncomplementarity conditions (i.e., women low in IO working with a friendly confederate, women high in IO working with an unfriendly confederate) would experience the interpersonal situation more negatively compared to women in complementarity conditions. Study results confirmed this prediction, with noncomplementarity in IO resulting in greater physiological reactivity, greater likelihood to attempt nonverbally to restore complementarity, more partner-related thoughts, and a reduced desire to seek out attention compared to women working in complementarity conditions. Implications for research on IO as a person variable are discussed.

On Linear Fractional Programming Problem and its Computation Using a Neural Network Model

In this paper we consider linear fractional programming problem and look at its linear complementarity formulation. In the literature, uniqueness of solution of a linear fractional programming problem is characterized through strong quasiconvexity. We present another characterization of uniqueness through complementarity approach and show that the solution set of a fractional programming problem is convex. Finally we formulate the complementarity condition as a set of dynamical equations and prove certain results involving the neural network model. A computational experience is also reported.

Piecewise linear NCP function for QP free feasible method

In this paper, a QP-free feasible method with piecewise NCP functions is proposed for nonlinear inequality constrained optimization problems. The new NCP functions are piecewise linear-rational, regular pseudo-smooth and have nice properties. This method is based on the solutions of linear systems of equation reformulation of KKT optimality conditions, by using the piecewise NCP functions. This method is implementable and globally convergent without assuming the strict complementarity condition, the isolatedness of accumulation points. Furthermore, the gradients of active constraints are not requested to be linearly independent. The submatrix which may be obtained by quasi-Newton methods, is not requested to be uniformly positive definite. Preliminary numerical results indicate that this new QP-free method is quite promising.

Convergence analysis of the augmented Lagrangian method for nonlinear second-order cone optimization problems

The paper focuses on the convergence rate of the augmented Lagrangian method for nonlinear second-order cone optimization problems. Under a set of assumptions of sufficient conditions, including the componentwise strict complementarity condition, the constraint nondegeneracy condition and the second-order sufficient condition, we first study some properties of the augmented Lagrangian and then show that the rate of local convergence of the augmented Lagrangian method is proportional to 1 / τ , where the penalty parameter τ is not less than a threshold τ ˆ > 0 .

Global Convergence of a Smooth Approximation Method for Mathematical Programs with Complementarity Constraints

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.

Leader-Follower Equilibria for Electric Power and NO x Allowances Markets

This paper investigates the ability of the largest producer in an electricity market to manipulate both the electricity and emission allowances markets to its advantage. A Stackelberg game to analyze this situation is constructed in which the largest firm plays the role of the leader, while the medium-sized firms are treated as Cournot followers with price-taking fringes that behave competitively in both markets. Since there is no explicit representation of the best-reply function for each follower, this Stackelberg game is formulated as a large-scale mathematical program with equilibrium constraints. The best-reply functions are implicitly represented by a set of nonlinear complementarity conditions. Analysis of the computed solution for the Pennsylvania–New Jersey–Maryland electricity market shows that the leader can gain substantial profits by withholding allowances and driving up NO x allowance costs for rival producers. The allowances price is higher than the corresponding price in the Nash–Cournot case, although the electricity prices are essentially the same.

Global Optimality Conditions for Discrete and Nonconvex Optimization—With Applications to Lagrangian Heuristics and Column Generation

The well-known and established global optimality conditions based on the Lagrangian formulation of an optimization problem are consistent if and only if the duality gap is zero. We develop a set of global optimality conditions that are structurally similar but are consistent for any size of the duality gap. This system characterizes a primal–dual optimal solution by means of primal and dual feasibility, primal Lagrangian ε-optimality, and, in the presence of inequality constraints, a relaxed complementarity condition analogously called δ-complementarity. The total size ε + δ of those two perturbations equals the size of the duality gap at an optimal solution. Further, the characterization is equivalent to a near-saddle point condition which generalizes the classic saddle point characterization of a primal–dual optimal solution in convex programming. The system developed can be used to explain, to a large degree, when and why Lagrangian heuristics for discrete optimization are successful in reaching near-optimal solutions. Further, experiments on a set-covering problem illustrate how the new optimality conditions can be utilized as a foundation for the construction of new Lagrangian heuristics. Finally, we outline possible uses of the optimality conditions in column generation algorithms and in the construction of core problems.

Partial Augmented Lagrangian Method and Mathematical Programs with Complementarity Constraints

In this paper, we apply a partial augmented Lagrangian method to mathematical programs with complementarity constraints (MPCC). Specifically, only the complementarity constraints are incorporated into the objective function of the augmented Lagrangian problem while the other constraints of the original MPCC are retained as constraints in the augmented Lagrangian problem. We show that the limit point of a sequence of points that satisfy second-order necessary conditions of the partial augmented Lagrangian problems is a strongly stationary point (hence a B-stationary point) of the original MPCC if the limit point is feasible to MPCC, the linear independence constraint qualification for MPCC and the upper level strict complementarity condition hold at the limit point. Furthermore, this limit point also satisfies a second-order necessary optimality condition of MPCC. Numerical experiments are done to test the computational performances of several methods for MPCC proposed in the literature.

A Strongly and Superlinearly Convergent SQP Algorithm for Optimization Problems with Linear Complementarity Constraints

This paper discusses a kind of optimization problem with linear complementarity constraints, and presents a sequential quadratic programming (SQP) algorithm for solving a stationary point of the problem. The algorithm is a modification of the SQP algorithm proposed by Fukushima et al. [Computational Optimization and Applications, 10 (1998), 5-34], and is based on a reformulation of complementarity condition as a system of linear equations. At each iteration, one quadratic programming and one system of equations needs to be solved, and a curve search is used to yield the step size. Under some appropriate assumptions, including the lower-level strict complementarity, but without the upper-level strict complementarity for the inequality constraints, the algorithm is proved to possess strong convergence and superlinear convergence. Some preliminary numerical results are reported.

A general MPCC model and its solution algorithm for continuous network design problem

This paper formulates the continuous network design problem as a mathematical program with complementarity constraints (MPCC), with the upper level a nonlinear programming problem and the lower level a nonlinear complementarity problem. Unlike in most previous studies, the proposed framework is more general, in which both symmetric and asymmetric user equilibria can be captured. By applying the complementarity slackness condition of the lower-level problem, the original bilevel formulation can be converted into a single-level and smooth nonlinear programming problem. In order to solve the problem, a relaxation scheme is applied by progressively restricting the complementarity condition, which has been proven to be a rigorous approach under certain conditions. The model and solution algorithm are tested for well-known network design problems and promising results are shown.

An Oligopolistic Model of an Integrated Market for Energy and Spinning Reserve

In this paper, a model for oligopolistic competition in electricity markets is presented. Most previous proposed models have been static and focused only on the energy market incentives for strategic behavior. In contrast, in this paper, a multiperiod market for energy and spinning reserve (SR) is considered. By including such factors, the competition among participants is modeled with more realism. Competition in the energy market is modeled by means of conjectured supply functions, while conjectured reserve-price response functions are used to consider the generators' ability to alter the SR prices. The resulting equilibrium problem is modeled in terms of complementarity conditions. Based upon a complementarity model, the opportunity cost between the energy and SR markets is derived for oligopolistic markets. The proposed model is illustrated by a six-node network using a dc approximation.

Shirking and Squandering in Sharing Games

In a sharing game the players' choices yield a revenue, each player's choice carries a cost, and a player's payoff is a portion of the revenue minus the player's cost. Such games are appealingly simple devices for partially aligning individual incentives with organizational goals, but their equilibria may be inefficient, i.e., at an equilibrium the surplus (revenue minus the sum of the costs) may not be maximal. Sharing games as a general class have not been well studied. We start a general theory of sharing games by going beyond the common economic setting, where strategy sets are continua and cost and revenue functions are smooth. We include games in which some of a player's strategies are equally costly, and revenue changes when he switches from one of them to another. We consider several large classes of reward functions, including nondecreasing residual (NDR) functions, in which residual (revenue minus rewards) does not drop when revenue increases. That class includes budget-balancing functions, where residual is always zero. To focus the discussion, we examine a ``Folk Claim", which asserts that at every inefficient equilibrium shirking, in some sense, occurs. We show that in NDR games a complementarity condition indeed insures that no one squanders at equilibrium (spends more than at an efficient profile). But when we drop complementarity, the situation changes sharply, and there are games with compelling equilibria, at which some players squander. The shirking/squandering distinction is particularly important in tracing the effect of technical improvement on the surplus shortfall at a sharing game's equilibrium. The paper also obtains conditions for existence of (pure-strategy) equilibria and finds, in particular, that every finite game in which rewards are linearly related has an equilibrium.

Organising cooperative institutional forms in knowledge transfer across borders a transaction cost approach to comparative firm performance

Organising cooperative institutional forms such as alliances, mergers, joint ventures and acquisitions on the basis of their comparative performance is critical for firms dealing in make or buy decisions. The decision may appear more significant if managers decide by comparing performance aspects of production activities of alternative cooperative forms. This study presents a less rigid view of the framework of transaction cost approach to develop its implications for organising cooperative governance forms. This framework adapts 'capability' as one of the transactional dimensions where transaction costs could be minimised within the exchange conditions of complementarity between the partners. This transactional dimension is conceptualised to minimise the risks of moral hazards, as the framework further translates the transactional dimension into complementary capabilities and complementary investments. Thus, the relationship between capability, complementary capability and complementary investments is conceptualised as a key performance measure of the cooperative relationship development process.

A Sequential Smooth Penalization Approach to Mathematical Programs with Complementarity Constraints

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.

Existence of Solutions for a Class of Impact Problems Without Viscosity

In this paper we consider dynamic frictionless impact problems of elastic materials formulated in abstract settings. The contact conditions for the impact problem are Signorini-type complementarity conditions. Using time discretization and Galerkin approximation, we investigate the convergence of numerical fully discrete trajectories to a solution of the continuous-time problem. In this way we establish the existence of solutions for a class of impact problems, some of which have been previously studied, while others have not. Most of the impact problems to which this theory can be applied are "thick" obstacle problems, although it can also be applied to a number of boundary or "thin" obstacle problems. The crucial assumption for the theory is that the cone of possible contact forces satisfies a strong pointedness condition, which can usually be related to a Sobolev embedding condition.

Balanced robust regulation of a magnetic levitation system

In this brief, we consider the problem of disturbance suppression for a magnetic levitation system in presence of physical uncertainties. The problem is cast as a nonlinear regulation problem in presence of input constraint and an internal model-based regulator is designed. A control law which does not rely upon complementarity or relaxed complementarity conditions is proposed and compared, from an energetic viewpoint, with complementary control strategies usually presented in literature. Simulation results are presented to confirm the effectiveness of the design.

Optimization of a Long-Short Portfolio under Nonconvex Transaction Cost

The purpose of this paper is to propose a practical branch and bound algorithm for solving a class of long-short portfolio optimization problem with concave and d.c. transaction cost and complementarity conditions on the variables. We will show that this algorithm can solve a problem of practical size and that the long-short strategy leads to a portfolio with significantly better risk-return structure compared with standard purchase only portfolio both in terms of ex-ante and ex-post performance.

Comparative cheap talk

When are comparative statements credible? We show that simple complementarity conditions ensure that an expert with private information about multiple issues can credibly rank the issues for a decision maker. By restricting the expert's ability to exaggerate, multidimensional cheap talk of this form permits communication when it would not be credible in a single dimension. The communication gains can be substantial with even a couple of dimensions, and the complete ranking is asymptotically equivalent to full revelation as the number of issues becomes large. Nevertheless, partial rankings are sometimes more credible and/or more profitable for the expert than the complete ranking. Comparative cheap talk is robust to asymmetries that are not too large. Consequently, for sufficiently many independent issues, there are always some issues sufficiently symmetric to permit comparative cheap talk.

The penalty interior-point method fails to converge†

Equilibrium equations in the form of complementarity conditions often appear as constraints in optimization problems. Problems of this type are commonly referred to as mathematical programs with complementarity constraints (MPCCs). A popular method for solving MPCCs is the penalty interior-point algorithm (PIPA). This paper presents an example for which PIPA converges to a nonstationary point, providing a counterexample to the established theory. The reasons for this adverse behavior are discussed. †Preprint ANL/MCS-P1091-0903.