Weak sharp minima revisited, part II: application to linear regularity and error bounds

The notion of weak sharp minima is an important tool in the analysis of the perturbation behavior of certain classes of optimization problems as well as in the convergence analysis of algorithms designed to solve these problems. It has been studied extensively by several authors. This paper is the second of a series on this subject where the basic results on weak sharp minima in Part I are applied to a number of important problems in convex programming. In Part II we study applications to the linear regularity and bounded linear regularity of a finite collection of convex sets as well as global error bounds in convex programming. We obtain both new results and reproduce several existing results from a fresh perspective. We dedicate this paper to our friend and mentor Terry Rockafellar on the occasion of his 70th birthday. He has been our guide in mathematics as well as in the backcountry and waterways of the Olympic and Cascade mountains. Research supported in part by the National Science Foundation Grant DMS-0203175.     

Strong Fermat Rules for Constrained Set-Valued Optimization Problems on Banach Spaces

In this paper, we propose two kinds of optimality concepts, called the sharp minima and the weak sharp minima, for a constrained set-valued optimization problem. Subsequently, we extend the Fermat rules for the local minima of the constrained set-valued optimization problem to the sharp minima and the weak sharp minima in Banach spaces or Asplund spaces, by means of the Mordukhovich generalized differentiation and the normal cone. As applications, we investigate the generalized inequality systems with constraints, and get some characterizations of error bounds for the constrained generalized inequality systems in convex and nonconvex cases.     

Slopes,Error Bounds and Weak Sharp Pareto Minima of a Vector-Valued Map

In this paper, we provide a detailed study of the upper and lower slopes of a vector-valued map recently introduced by Bednarczuk and Kruger. We show that these slopes enjoy most properties of the strong slope of a scalar-valued function and can be explicitly computed or estimated in the convex, strictly differentiable, linear cases. As applications, we obtain error bounds for lower level sets (in particular, a Hoffman-type error bound for a system of linear inequalities in the infinite-dimensional space setting, existence of weak sharp Pareto minima) and sufficient conditions for Pareto minima.     

The Equivalence of Three Types of Error Bounds for Weakly and Approximately Convex Functions

Journal of Optimization Theory and Applications - We start by establishing the equivalence of three types of error bounds: weak sharp minima, level-set subdifferential error bounds and...     

Generalized weak sharp minima in cone-constrained convex optimization on Hadamard manifolds

In this paper, a convex optimization problem with cone constraint (for short, CPC) is introduced and studied on Hadamard manifolds. Some criteria and characterizations for the solution set to be a set of generalized global weak sharp minima, generalized local weak sharp minima and generalized bounded weak sharp minima for (CPC) are derived on Hadamard manifolds.     

GLOBAL WEAK SHARP MINIMA AND COMPLETENESS OF METRIC SPACE

A sufficient condition on the existence of a global weak sharp minima for general function in metric space is established. A characterization for convex function to have global weak sharp minima is also presented, which generalized Burke and Ferris‘ result to infinite dimensional space. A characterization of the completeness of a metric space is given by the existence of global weak sharp minima.     

Error bounds for set inclusions

A variant of Robinson-Ursescu Theorem is given in normed spaces. Several error bound theorems for convex inclusions are proved and in particular a positive answer to Li and Singer's conjecture is given under weaker assumption than the assumption required in their conjecture. Perturbation error bounds are also studied. As applications, we study error bounds for convex inequality systems.     

Finite termination of the proximal point algorithm in Banach spaces

In this paper, we show that the convex optimization problem can be solved by the proximal point algorithm in a finite number of steps under the assumption that the solution set is a set of weak sharp minima.     

Global weak sharp minima for convex (semi-)infinite optimization problems

We mainly consider global weak sharp minima for convex infinite and semi-infinite optimization problems (CIP). In terms of the normal cone, subdifferential and directional derivative, we provide several characterizations for (CIP) to have global weak sharp minimum property.     

Generalized weak sharp minima in cone-constrained convex optimization with applications

In this paper, we consider convex optimization problems with cone constraints (CPC in short). We study generalized weak sharp minima properties for (CPC) in the Banach space and Hilbert space settings, respectively. Some criteria and characterizations for the solution set to be a set of generalized weak sharp minima for (CPC) are derived. As an application, we propose an algorithm for (CPC) in the Hilbert space setting. Convergence analysis of this algorithm is given.     

New characterizations of weak sharp minima

In this paper, we study the concept of weak sharp minima by using conjugate functions. Not only some well-known results can be obtained in this unified way, but also new characterizations are developed. Finally, under rather weak conditions, we establish the finite termination property for convex programming and variational inequality problem, respectively.     

Gap functions and error bounds for nonsmooth convex vector optimization problem

Abstract

In this article, our main aim is to develop gap functions and error bounds for a (non-smooth) convex vector optimization problem. We show that by focusing on convexity we are able to quite efficiently compute the gap functions and try to gain insight about the structure of set of weak Pareto minimizers by viewing its graph. We will discuss several properties of gap functions and develop error bounds when the data are strongly convex. We also compare our results with some recent results on weak vector variational inequalities with set-valued maps, and also argue as to why we focus on the convex case.     

On convergence of the Gauss-Newton method for convex composite optimization

The local quadratic convergence of the Gauss-Newton method for convex composite optimization f=h∘F is established for any convex function h with the minima set C, extending Burke and Ferris’ results in the case when C is a set of weak sharp minima for h. Received: July 24, 1998 / Accepted: November 29, 2000?Published online September 3, 2001     

Mean lattice point discrepancy bounds,II: Convex domains in the plane

We consider planar curved strictly convex domains with very weak (or no) smoothness assumptions on the boundary and prove sharp bounds for square-functions associated to the lattice point discrepancy. Research supported in part by NSF grants.     

Sharp and simple bounds for the Erlang delay and loss formulae

We prove some simple and sharp lower and upper bounds for the Erlang delay and loss formulae and for the number of servers that invert the Erlang delay and loss formulae. We also suggest simple and sharp approximations for the number of servers that invert the Erlang delay and loss formulae. We illustrate the importance of these bounds by using them to establish convexity proofs. We show that the probability that the M/M/s queue is empty is a decreasing and convex function of the traffic intensity. We also give a very short proof to show that the Erlang delay formula is convex in the traffic intensity when the number of servers is held constant. The complete proof of this classical result has never been published. We also give a very short proof to show that the Erlang delay formula is a convex function of the (positive integer) number of servers. One of our results is then used to get a sharp bound to the Flow Assignment Problem.     

A lower bound on the number of sharp shadow-boundaries of convex polytopes

We give the lower bound on the number of sharp shadow-boundaries of convexd-polytopes (or unbounded convex polytopal sets) withn facets. The polytopes (sets) attaining these bounds are characterized. Additionally, our results will be transferred to the dual theory.The research work of the first author was (partially) supported by Hungarian National Foundation for Scientific Research, grant no. 1812.     

Notes on lattice points of zonotopes and lattice-face polytopes

Minkowski’s second theorem on successive minima gives an upper bound on the volume of a convex body in terms of its successive minima. We study the problem to generalize Minkowski’s bound by replacing the volume by the lattice point enumerator of a convex body. In this context we are interested in bounds on the coefficients of Ehrhart polynomials of lattice polytopes via the successive minima. Our results for lattice zonotopes and lattice-face polytopes imply, in particular, that for 0-symmetric lattice-face polytopes and lattice parallelepipeds the volume can be replaced by the lattice point enumerator.     

Partially finite convex programming,Part II: Explicit lattice models

In Part I of this work we derived a duality theorem for partially finite convex programs, problems for which the standard Slater condition fails almost invariably. Our result depended on a constraint qualification involving the notion ofquasi relative interior. The derivation of the primal solution from a dual solution depended on the differentiability of the dual objective function: the differentiability of various convex functions in lattices was considered at the end of Part I. In Part II we shall apply our results to a number of more concrete problems, including variants of semi-infinite linear programming,L ¹ approximation, constrained approximation and interpolation, spectral estimation, semi-infinite transportation problems and the generalized market area problem of Lowe and Hurter (1976). As in Part I, we shall use lattice notation extensively, but, as we illustrated there, in concrete examples lattice-theoretic ideas can be avoided, if preferred, by direct calculation.     

A unified approach to error bounds for structured convex optimization problems

Error bounds, which refer to inequalities that bound the distance of vectors in a test set to a given set by a residual function, have proven to be extremely useful in analyzing the convergence rates of a host of iterative methods for solving optimization problems. In this paper, we present a new framework for establishing error bounds for a class of structured convex optimization problems, in which the objective function is the sum of a smooth convex function and a general closed proper convex function. Such a class encapsulates not only fairly general constrained minimization problems but also various regularized loss minimization formulations in machine learning, signal processing, and statistics. Using our framework, we show that a number of existing error bound results can be recovered in a unified and transparent manner. To further demonstrate the power of our framework, we apply it to a class of nuclear-norm regularized loss minimization problems and establish a new error bound for this class under a strict complementarity-type regularity condition. We then complement this result by constructing an example to show that the said error bound could fail to hold without the regularity condition. We believe that our approach will find further applications in the study of error bounds for structured convex optimization problems.     

Forward error analysis of Gaussian elimination

Summary Part I of this work deals with the forward error analysis of Gaussian elimination for general linear algebraic systems. The error analysis is based on a linearization method which determines first order approximations of the absolute errors exactly. Superposition and cancellation of error effects, structure and sparsity of the coefficient matrices are completely taken into account by this method. The most important results of the paper are new condition numbers and associated optimal component-wise error and residual estimates for the solutions of linear algebraic systems under data perturbations and perturbations by rounding erros in the arithmetic floating-point operations. The estimates do not use vector or matrix norms. The relative data and rounding condition numbers as well as the associated backward and residual stability constants are scaling-invariant. The condition numbers can be computed approximately from the input data, the intermediate results, and the solution of the linear system. Numerical examples show that by these means realistic bounds of the errors and the residuals of approximate solutions can be obtained. Using the forward error analysis, also typical results of backward error analysis are deduced. Stability theorems and a priori error estimates for special classes of linear systems are proved in Part II of this work.