Violator Spaces: Structure and Algorithms

Sharir and Welzl introduced an abstract framework for optimization problems, called LP-type problems or also generalized linear programming problems, which proved useful in algorithm design. We define a new, and as we believe, simpler and more natural framework: violator spaces, which constitute a proper generalization of LP-type problems. We show that Clarkson's randomized algorithms for low-dimensional linear programming work in the context of violator spaces. For example, in this way we obtain the fastest known algorithm for the P-matrix generalized linear complementarity problem with a constant number of blocks. We also give two new characterizations of LP-type problems: they are equivalent to acyclic violator spaces, as well as to concrete LP-type problems (informally, the constraints in a concrete LP-type problem are subsets of a linearly ordered ground set, and the value of a set of constraints is the minimum of its intersection).Comment: 28 pages, 5 figures, extended abstract was presented at ESA 2006; author spelling fixe

Necessary and sufficient conditions for unique solvability of absolute value equations: A Survey

In this survey paper, we focus on the necessary and sufficient conditions for the unique solvability and unsolvability of the absolute value equations (AVEs) during the last twenty years (2004 to 2023). We discussed unique solvability conditions for various types of AVEs like standard absolute value equation (AVE), Generalized AVE (GAVE), New generalized AVE (NGAVE), Triple AVE (TAVE) and a class of NGAVE based on interval matrix, P-matrix, singular value conditions, spectral radius and $\mathcal{W}$-property. Based on the unique solution of AVEs, we also discussed unique solvability conditions for linear complementarity problems (LCP) and horizontal linear complementarity problems (HLCP)

Unique Sink Orientations of Grids

We introduce unique sink orientations of grids as digraph models for many well-studied problems, including linear programming over products of simplices, generalized linear complementarity problems over P-matrices (PGLCP), and simple stochastic games. We investigate the combinatorial structure of such orientations and develop randomized algorithms for finding the sink. We show that the orientations arising from PGLCP satisfy the Holt-Klee condition known to hold for polytope digraphs, and we give the first expected linear-time algorithms for solving PGLCP with a fixed number of block

Block Principal Pivoting Algorithm for VGLCP: A Block Principal Pivoting Algorithm for the Vertical Generalized Linear Complementarity Problem (VGLCP) with a Vertical Block P-matrix

A block principal pivoting algorithm for the vertical generalized linear complementarity problem (VGLCP) associated with a vertical block P-matrix is provided. A preliminary result shows that it is very efficient when applied to large scaled problems as it reduces the number of iterations needed to solve the VGLCP when compared to other direct algorithms

Charactarizations of Linear Suboptimality for Mathematical Programs with Equilibrium Constraints

The paper is devoted to the study of a new notion of linear suboptimality in constrained mathematical programming. This concept is different from conventional notions of solutions to optimization-related problems, while seems to be natural and significant from the viewpoint of modern variational analysis and applications. In contrast to standard notions, it admits complete characterizations via appropriate constructions of generalized differentiation in nonconvex settings. In this paper we mainly focus on various classes of mathematical programs with equilibrium constraints (MPECs), whose principal role has been well recognized in optimization theory and its applications. Based on robust generalized differential calculus, we derive new results giving pointwise necessary and sufficient conditions for linear suboptimality in general MPECs and its important specifications involving variational and quasi variational inequalities, implicit complementarity problems, etc

A generalized complementarity approach to solving real option problems

This article provides a unified framework for analyzing a wide variety of real option problems. These problems include the frequently studied, simple real option problems, as described in More specifically, all of the present real option problems can be directly solved by reformulating their optimality condition as a dynamical system of generalized linear complementarity problems (GLCPs). This enables us to develop an efficient and robust algorithm for solving a broad range of real option problems in a unified manner, exploiting recent advances in the theory of complementarity problems. JEL classification: C61; C63; D92; E22; G3