Characterizations of long-run producer optima and the short-runapproach to long-run market equilibrium: a general theory withapplications to peak-load pricing

This is a new formal framework for the theory of competitive equilibrium and its applications.Our "short-run approach" means the calculation of long-run producer optimaand general equilibria from the short-run solutions to the producer's profit maximizationprogramme and its dual. The marginal interpretation of the dual solution means that itcan be used to value the capital and other fixed inputs, whose levels are then adjustedaccordingly (where possible). But short-run profit can be a nondifferentiable function ofthe fixed quantities, and the short-run cost is nondifferentiable whenever there is a rigidcapacity constraint. Nondifferentiability of the optimal value requires the introductionof nonsmooth calculus into equilibrium analysis, and subdifferential generalizations ofsmooth-calculus results of microeconomics are given, including the key Wong-Viner EnvelopeTheorem. This resolves long-standing discrepancies between "textbook theory"and industrial experience. The other tool employed to characterise long-run produceroptima is a primal-dual pair of programmes. Both marginalist and programming characterizationsof producer optima are given in a taxonomy of seventeen equivalent systemsof conditions. When the technology is described by production sets, the most usefulsystem for the short-run approach is that using the short-run profit programme andits dual. This programme pair is employed to set up a formal framework for long-rungeneral-equilibrium pricing of a range of commodities with joint costs of production.This gives a practical method that finds the short-run general equilibrium en route tothe long-run equilibrium, exploiting the operating policies and plant valuations that mustbe determined anyway. These critical short-run solutions have relatively simple formsthat can greatly ease the fixed-point problem of solving for equilibrium, as is shownon an electricity pricing example. Applicable criteria are given for the existence of theshort-run solutions and for the absence of a duality gap. The general analysis is speltout for technologies with conditionally fixed coefficients, a concept extending that of thefixed-coefficients production function to the case of multiple outputs. The short-run approachis applied to the peak-load pricing of electricity generated by thermal, hydro andpumped-storage plants. This gives, for the first time, a sound method of valuing thefixed assets-in this case, river flows and the sites suitable for reservoirs.general equilibrium, fixed-input valuation, nondifferentiable joint costs,Wong-Viner Envelope Theorem, public utility pricing

Stability of the Duality Gap in Linear Optimization

In this paper we consider the duality gap function g that measures the difference between the optimal values of the primal problem and of the dual problem in linear programming and in linear semi-infinite programming. We analyze its behavior when the data defining these problems may be perturbed, considering seven different scenarios. In particular we find some stability results by proving that, under mild conditions, either the duality gap of the perturbed problems is zero or + ∞ around the given data, or g has an infinite jump at it. We also give conditions guaranteeing that those data providing a finite duality gap are limits of sequences of data providing zero duality gap for sufficiently small perturbations, which is a generic result.This research was partially supported by MINECO of Spain and FEDER of EU, Grant MTM2014-59179-C2-01 and SECTyP-UNCuyo Res. 4540/13-R

Zero-Convex Functions, Perturbation Resilience, and Subgradient Projections for Feasibility-Seeking Methods

The convex feasibility problem (CFP) is at the core of the modeling of many problems in various areas of science. Subgradient projection methods are important tools for solving the CFP because they enable the use of subgradient calculations instead of orthogonal projections onto the individual sets of the problem. Working in a real Hilbert space, we show that the sequential subgradient projection method is perturbation resilient. By this we mean that under appropriate conditions the sequence generated by the method converges weakly, and sometimes also strongly, to a point in the intersection of the given subsets of the feasibility problem, despite certain perturbations which are allowed in each iterative step. Unlike previous works on solving the convex feasibility problem, the involved functions, which induce the feasibility problem's subsets, need not be convex. Instead, we allow them to belong to a wider and richer class of functions satisfying a weaker condition, that we call "zero-convexity". This class, which is introduced and discussed here, holds a promise to solve optimization problems in various areas, especially in non-smooth and non-convex optimization. The relevance of this study to approximate minimization and to the recent superiorization methodology for constrained optimization is explained.Comment: Mathematical Programming Series A, accepted for publicatio

On the local stability of semidefinite relaxations

We consider a parametric family of quadratically constrained quadratic programs (QCQP) and their associated semidefinite programming (SDP) relaxations. Given a nominal value of the parameter at which the SDP relaxation is exact, we study conditions (and quantitative bounds) under which the relaxation will continue to be exact as the parameter moves in a neighborhood around the nominal value. Our framework captures a wide array of statistical estimation problems including tensor principal component analysis, rotation synchronization, orthogonal Procrustes, camera triangulation and resectioning, essential matrix estimation, system identification, and approximate GCD. Our results can also be used to analyze the stability of SOS relaxations of general polynomial optimization problems.Comment: 23 pages, 3 figure

A nonlinear Lagrangian approach to constrained optimization problems

Author name used in this publication: Yang, X. Q.2000-2001 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishe