Some properties of regularized and penalized nonlinear programming formulations of mathematical programs with equilibrium constraints (MPECs) are described. The focus is on the properties of these formulations near a local solution of the MPEC at which strong stationarity and a second-order sufficient condition are satisfied. In the regularized formulations, the complementarity condition is replaced by … Read more
A classical theorem by Block and Levin says that certain variants of the relaxation method for solving systems of linear inequalities produce bounded sequences of intermediate solutions even when running on inconsistent input data. Using a new approach, we prove a more general version of this result and answer an old open problem of quantifying … Read more
It has been known for almost 50 years that the discrete l_1 approximation problem can be solved effectively by linear programming. However, improved algorithms involve a step which can be interpreted as a line search, and which is not part of the standard LP solution procedures. l_1 provides the simplest example of a class of … Read more
It is well known that the optimal values of a linear programming problem and its dual are equal to each other if at least one of these problems is feasible. It is also well known that for linear conic problems this property of “no duality gap” may not hold. It is then natural to ask … Read more
This paper draws from our experience in developing the AMPL modeling language, to show where numerical issues have been crucial to modeling language design and where modeling language advances have strongly influenced the design of solvers. CitationProceedings of the 20th Biennial Conference on Numerical Analysis, Dundee, Scotland, D.F. Griffiths and G.A. Watson, eds., University of … Read more
We study the convex hull of the feasible set of the semi-continuous knapsack problem, in which the variables belong to the union of two intervals. Besides being important in its own right, the semi-continuous knapsack problem is a relaxation of general mixed-integer programming. We show how strong inequalities valid for the semi-continuous knapsack polyhedron can … Read more
We consider an augmented Lagrangian algorithm for minimizing a convex quadratic function subject to linear inequality constraints. Linear optimization is an important particular instance of this problem. We show that, provided the augmentation parameter is large enough, the constraint value converges {\em globally\/} linearly to zero. This property is viewed as a consequence of the … Read more
Several Multi-Criteria-Decision-Making methodologies assume the existence of weights associated with the different criteria, reflecting their relative importance. One of the most popular ways to infer such weights is the Analytic Hierarchy Process, which constructs first a matrix of pairwise comparisons, from which weights are derived following one out of many existing procedures, such as the … Read more
We consider the integer program P -> max {c’x | Ax=y; x\in N^n }. Using the generating function of an associated counting problem, and a generalized residue formula of Brion and Vergne, we explicitly relate P with its continuous linear programming (LP) analogue and provide a characterization of its optimal value. In particular, dual variables … Read more
Let f be a lower semi-continuous convex function in a Euclidean space, finite or infinite dimensional. The gradient differential inclusion is a generalized differential equation which requires that -x'(t) be in the subgradient of f at x. It is the continuous versions of the gradient descent method for minimizing f in case f is differentiable, … Read more