scispace - formally typeset
Search or ask a question
Book

Linear complementarity, linear and nonlinear programming

01 Jan 1988-
About: The article was published on 1988-01-01 and is currently open access. It has received 1012 citations till now. The article focuses on the topics: Mixed complementarity problem & Complementarity theory.
Citations
More filters
Posted Content
TL;DR: A real symmetric n times n matrix is called copositive if the corresponding quadratic form is non-negative on the closed first orthant of the matrix as discussed by the authors.
Abstract: A real symmetric n times n matrix is called copositive if the corresponding quadratic form is non-negative on the closed first orthant. If the matrix fails to be copositive there exists some non-negative certificate for which the quadratic form is negative. Due to the scaling property, we can find such certificates in every neighborhood of the origin but their properties depend on the matrix of course and are hard to describe. If it is an integer matrix however, we are guaranteed certificates of a complexity that is at most a constant times the binary encoding length of the matrix raised to the power 3/2.

Cites background from "Linear complementarity, linear and ..."

  • ...If not, we proceed as follows: Assume s is no extreme point, then there exist distinct feasible solutions t((1)), t((2)) and α ∈ (0, 1) s....

    [...]

  • ...the non-negativity of t((1)), t((2)) and α ∈ (0, 1) together imply that s j = 0 forces t (1) j = t (2) j = 0....

    [...]

  • ...(12) With x̄ being an optimal solution to problem (2) and the corresponding vectors ū, v̄, ȳ ∈ R≥0 defined as in the foregoing lemma, we know that...

    [...]

  • ...= α · ( (x((1)))Mx((1)) ) + (1− α) · ( (x((2)))Mx((2)) )...

    [...]

  • ...For an optimal solution x̄ to (2), there exist vectors ȳ, ū, v̄ ∈ R≥0 such that ( ū v̄ ) − ( M I −I 0 ) · ( x̄ ȳ ) = ( 0 e ) and (3)...

    [...]

Journal ArticleDOI
TL;DR: In this article , an inexact-Uzawa solver for embedded linear model predictive control (MPC) is proposed, which falls into the general framework of first-order primal-dual methods but employs both proximal-point and matrix splitting schemes to derive a numerically robust algorithm.
Abstract: In this letter, we propose an inexact-Uzawa solver for embedded linear model predictive control (MPC). The inexact-Uzawa algorithm falls into the general framework of first-order primal-dual methods but employs both proximal-point and matrix splitting schemes to derive a numerically robust algorithm with $\mathcal {O}$ ( $1/k$ ) convergence rate in the primal-dual gap to some saddle-point solution where $k$ is the iteration count. Numerical MPC example shows the efficiency and the ease of implementation of the algorithm as compared to other related methods in the literature.
Posted Content
TL;DR: In this article, the authors study an unbalanced multisectoral growth path in which cost-minimizing producers are subjected to a system of differential interest rates and show that the existence of competitive equilibrium is ensured if interest rate differentials are "in tune" with physical productivity differentials.
Abstract: The paper studies an unbalanced multisectoral growth path in which cost-minimizing producers are subjected to a system of differential interest rates. We ask whether such differentials in capital charges are compatible with the existence of competitive equilibrium. In Part I, we study the price-quantity equilibrium for given final demand and a given level of interest rates. We find that existence is ensured if interest-rate differentials are "in tune" with physical productivity differentials. Our proof is constructive; we show that an equilibrium solution can be computed by applying Lemke's complementary pivoting algorithm to the data of the problem. In Part II, we study how the level of interest rates and the level and structure of final demand adjust to changes in activity levels, prices and income. We find that the interest-rate differentials from Part I do not prevent the required equilibrium adjustments. This confirms our principal insight from Part I, namely that interest rate differentials are consistent with equilibrium as long as they are in tune with productivity differentials.
01 Jan 2011
TL;DR: A pivoting heuristic based on tabu search and its integration into an enumerative framework for solving the Linear Complementarity Problem (LCP) is described in this paper.
Abstract: This paper describes a pivoting heuristic based on tabu search and its integration into an enumerative framework for solving the Linear Complementarity Problem (LCP). The tabu pivoting heuristic works with basic solutions and performs pivot operations guided by two indicators, one concerned with the satisfaction of the complementarity conditions and the other with the feasibility of the solution. It incorporates the concept of tabu search employing a strategy that avoids the repetition of recent moves. The heuristic ends when a solution to the LCP is found or after a specied number of iterations. In the latter case, an enumerative algorithm is applied which integrates the tabu pivoting heuristic within a branching framework. Computational experience on test problems is reported to highlight the eciency of the proposed methodology for solving the LCP.