Abstract. This paper gives an extensive documentation of applications of finite-dimensional nonlinear complementarity problems in engineering and equilibrium modeling. For most applications, we describe the problem briefly, state the defining equations of the model, and give functional expressions for the complementarity formulations. The goal of this documentation is threefold: (i) to summarize the essential applications of the nonlinear complementarity problem known to date, (ii) to provide a basis for the continued research on the nonlinear complementarity problem, and (iii) to supply a broad collection of realistic complementarity problems for use in algorithmic experimentation and other studies.

Lagrange multipliers used to be viewed as auxiliary variables introduced in a problem of constrained minimization in order to write first-order optimality conditions formally as a system of equations. Modern applications, with their emphasis on numerical methods and more complicated side conditions than equations, have demanded deeper understanding of the concept and how it fits into a larger theoretical picture. A major line of research has been the nonsmooth geometry of one-sided tangent and normal vectors to the set of points satisfying the given constraints. Another has been the game-theoretic role of multiplier vectors as solutions to a dual problem. Interpretations as generalized derivatives of the optimal value with respect to problem parameters have also been explored. Lagrange multipliers are now being seen as arising from a general rule for the subdifferentiation of a nonsmooth objective function which allows black-and-white constraints to be replaced by penalty expressions. This paper traces such themes in the current theory of Lagrange multipliers, providing along the way a freestanding exposition of basic nonsmooth analysis as motivated by and applied to this subject.

The origins and some motivational details of a collection of nonlinear mixed complementarity problems are given. This collection serves two purposes. Firstly, it gives a uniform basis for testing currently available and new algorithms for mixed complementarity problems. Function and Jacobian evaluations for the resulting problems are provided via a GAMS interface, making thorough testing of algorithms on practical complementarity problems possible. Secondly, it gives examples of how to formulate many popular problem formats as mixed complementarity problems and how to describe the resulting problems in GAMS format. We demonstrate the ease and power of formulating practical models in the MCP format. Given these examples, it is hoped that this collection will grow to include many problems that test complementarity algorithms more fully. The collection is available by anonymous ftp. Computational results using the PATH solver covering all of these problems are described. 1 Introduction R...

Abstract. Two fundamental classes of problems in large-scale linear and quadratic programming are described. Multistage problems covering a wide variety of models in dynamic programming and stochastic programming are represented in a new way. Strong properties of duality are revealed which support the development of iterative approximate techniques of solution in terms of saddlepoints. Optimality conditions are derived in a form that emphasizes the possibilities of decomposition.

Abstract. Many large-scale problems in dynamic and stochastic optimization can be modeled with extended linear-quadratic programming, which admits penalty terms and treats them through duality. In general the objective functions in such problems are only piecewise smooth and must be minimized or maximized relative to polyhedral sets of high dimensionality. This paper proposes a new class of numerical methods for “fully quadratic ” problems within this framework, which exhibit second-order nonsmoothness. These methods, combining the idea of finite-envelope representation with that of modified gradient projection, work with local structure in the primal and dual problems simultaneously, feeding information back and forth to trigger advantageous restarts. Versions resembling steepest descent methods and conjugate gradient methods are presented. When a positive threshold of ε-optimality is specified, both methods converge in a finite number of iterations. With threshold 0, it is shown under mild assumptions that the steepest descent version converges linearly, while the conjugate gradient version still has a finite termination property. The algorithms are designed to exploit features of primal and dual decomposability of the Lagrangian, which are typically available in a large-scale setting, and they are open to considerable parallelization. Key words. Extended linear-quadratic programming, large-scale numerical optimization, finite-envelope representation, gradient projection, primal-dual methods, steepest descent methods, conjugate gradient methods. AMS(MOS) subject classifications. 65K05, 65K10, 90C20 1. Introduction. A

We introduce Stochastic Mathematical Programs with Equilibrium Constraints (SMPEC), which generalize MPEC models to explicitly incorporate possible uncertainties in the problem data to obtain robust solutions to hierarchical problems. For this problem, we establish results on the existence of solutions, and on the convexity and directional differentiability of the implicit upper-level objective function, both for continuously and discretely distributed probability distributions. In so doing, we establish the links between SMPEC models and two-stage stochastic programs with recourse. We also introduce basic parallel iterative algorithms for discretely distributed SMPEC problems. 1 Introduction The present paper serves to introduce a framework for hierarchical decisionmaking under uncertainty. Hierarchical decision-making problems are encountered in a wide variety of domains in the engineering and experimental natural sciences, and in regional planning, management, and economics. These ...

An algorithm incorporating the logarithmic barrier into the Benders decomposition technique is proposed for solving two-stage stochastic programs. Basic properties concerning the existence and uniqueness of the solution and the underlying path are studied. When applied to problems with a finite number of scenarios, the algorithm is shown to converge globally and to run in polynomial-time. Key Words: Stochastic programming, Large-scale linear programming, Barrier function, Interior point methods, Benders decomposition, Complexity. Abbreviated Title: A log-barrier method with Benders decomposition AMS(MOS) subject classifications: 90C15, 90C05, 90C06, 90C60. 1 1. Introduction In this paper we propose an algorithm for solving two-stage stochastic programs, establish fundamental properties of the algorithm, and analyze the convergence. An example of a two-stage stochastic program is a production planning problem. The production and demand take place in the first and second periods, resp...

In this paper, stochastic programming problems are viewed as parametric programs with respect to the probability distributions of the random coefficients. General results on quantitative stability in parametric optimization are used to study distribution sensitivity of stochastic programs. For recourse and chance constrained models quantitative continuity results for optimal values and optimal solution sets are proved (with respect to suitable metrics on the space of probability distributions). The results are useful to study the effect of approximations and of incomplete information in stochastic programming.

This thesis is concerned with algorithms and software for the solution of the Mixed Complementarity Problem, or MCP. The MCP formulation is useful for expressing systems of nonlinear inequalities and equations; the complementarity allows boundary conditions be to specified in a succinct manner. Problems of this type occur in many branches of the sciences, including mathematics, engineering, economics, operations research, and computer science. The algorithm we propose for the solution of MCP is a Newton based method containing a novel application of a nonmonotone stabilization technique previously applied to methods for solving smooth systems of equalities and for unconstrained minimization. In order to apply this technique, we have adapted and extended the path construction technique of Ralph (1994), resulting in the PATH algorithm. We present a global convergence result for the PATH algorithm that generalizes similar results obtained in the smooth case. The PATH solver is a sophisticated implementation of this algorithm that makes use of the sparse basis updating package of MINOS 5.4. Due to the widespread use of algebraic modeling languages in the practice of operations research, economics, and other fields from which complementarity problems are drawn, we have developed a complementarity facility for both the GAMS and AMPL modeling languages, as well as software interface libraries to be used in hooking up a complementarity solver as a solution subsystem. These interface libraries provide the algorithm developer with a convenient and efficient means of developing and testing an algorithm, ...

While decisions frequently have uncertain consequences, optimal decision models often replace those uncertainties with averages or best estimates. Limited computational capability may have motivated this practice in the past. Recent computational advances have, however, greatly expanded the range of stochastic programs, optimal decision models with explicit consideration of uncertainties. This paper describes basic methodology in stochastic programming, recent developments in computation, and some practical application examples.