We consider an extension of the horizontal linear complementarity problem, which we call the extended linear complementarity problem (XLCP). With the aid of a natural bilinear program, we establish various properties of this extended complementarity problem; these include the convexity of the bilinear objective function under a monotonicity assumption, the polyhedrality of the solution set of a monotone XLCP, and an error bound result for a nondegenerate XLCP. We also present a finite, sequential linear programming algorithm for solving the nonmonotone XLCP.

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For the extended linear complementarity problem [11], we introduce and characterize column-sufficiency, row-sufficiency, and P-properties. These properties are then specialized to the vertical, horizontal, and mixed linear complementarity problems.

Abstract. During the last few decades, significant progress has been made in solving largescale finite-dimensional and semi-infinite linear programming problems. In contrast, little progress has been made in solving linear programs in infinite-dimensional spaces despite their importance as models in manufacturing and communication systems. Inspired by the research on separated continuous linear programs, we propose a new class of continuous linear programming problems that has a variety of important applications in communications, manufacturing, and urban traffic control. This class of continuous linear programs contains the separated continuous linear programs as a subclass. Using ideas from quadratic programming, we propose an efficient algorithm for solving large-scale problems in this new class under mild assumptions on the form of the problem data. We prove algorithmically the absence of a duality gap for this class of problems without any boundedness assumptions on the solution set. We show this class of problems admits piecewise constant optimal control when the optimal solution exists. We give conditions for the existence of an optimal solution. We also report computational results which illustrate that the new algorithm is effective in solving large-scale realistic problems (with several hundred continuous variables) arising in manufacturing systems.

We study the problem of solving a constrained system of nonlinear equations by a combination of the classical damped Newton method for (unconstrained) smooth equations and the recent interior point potential reduction methods for linear programs, linear and nonlinear complementarity problems. In general, constrained equations provide a unified formulation for many mathematical programming problems, including complementarity problems of various kinds and the Karush-Kuhn-Tucker systems of variational inequalities and nonlinear programs. Combining ideas from the damped Newton and interior point methods, we present an iterative algorithm for solving a constrained system of equations and investigate its convergence properties. Specialization of the algorithm and its convergence analysis to complementarity problems of various kinds and the Karush-Kuhn-Tucker systems of variational inequalities are discussed in detail. We also report the computational results of the implementation of the algo...

The extended linear complementarity problem (XLCP) has been introduced in a recent paper by Mangasarian and Pang. In the present research, minimization problems with simple bounds associated to this problem are defined. When the XLCP is solvable, their solutions are global minimizers of the associated problems. Sufficient conditions that guarantee that stationary points of the associated problems are solutions of the XLCP will be proved. These theoretical results support the conjecture that local methods for box constrained optimization applied to the associated problems could be efficient tools for solving the XLCP. Keywords. Complementarity, box constrained minimization. AMS: 90C33, 90C30 Department of Computer Science and Statistics, University of the State of S. Paulo (UNESP), C.P. 136, CEP 15054-000, S~ao Jos'e do Rio Preto-SP, Brazil. This author was supported by FAPESP (Grant 96-1552-0, 96/12503-0). E-mail: andreani@nimitz.dcce.ibilce.unesp.br y Department of Mathematics, I...

Abstract. We consider the extended linear complementarity problem (XLCP) introduced by Mangasarian and Pang [22], of which the horizontal and vertical linear complementarity problems are two special cases. We give some new sufficient conditions for every stationary point of the natural bilinear program associated with XLCP to be a solution of XLCP. We further propose some unconstrained and bound constrained reformulations for XLCP, and study the properties of their stationary points under assumptions similar to those for the bilinear program.

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sity function. By means of this approximation, linear and convex inequalities are converted into smooth, convex unconstrained minimization problems, the solution of which approximates the solution of the original problem to a high degree of accuracy for sufficiently small positive value of the smoothing param- eter fl. In the special case when a Slater constraint qualification is satisfied, an exact solution can be obtained for finite fl. Speedup over the linear/nonlinear programming package MINOS 5.4 was as high as 1142 times for linear inequali- ties of size 2000 x 1000, and 580 times for convex inequalities with 400 variables. Linear complementarity problems(LCPs) were treated by converting them into a system of smooth nonlinear equations and are solved by a quadratically con- vergent Newton method. For monotone LCPs with as many as 10,000 variables, the proposed approach was as much as 63 times faster than Lemke's method. Our smooth approach can also be used to solve nonlinear and mixed comple- menrarity problems (NCPs and MCPs) by converting them to classes of smooth parametric nonlinear equations. For any solvable NCP or MCP, existence of an arbitrarily accurate solution to the smooth nonlinear equation as well as the NCP or MCP, is established for sufficiently large value of a smoothing param- eter c = -l. An efficient smooth algorithm, based on the Newton-Armijo approach with an adjusted smoothing parameter, is also given and its global and local quadratic convergence is established. For NCPs, exact solutions of our smooth nonlinear equation for various values of the parameter c, generate an interior path, which is different from the central path for the interior point method. Computational results for 52 test problems compare favorably with those for another Ne...

The extended linear complementarity problem and its applications in the max-plus algebra ∗