A Fresh Variational-Analysis Look at the Positive Semidefinite Matrices World

International audienceEngineering sciences and applications of mathematics show unambiguously that positive semidefiniteness of matrices is the most important generalization of non-negative real num- bers. This notion of non-negativity for matrices has been well-studied in the literature; it has been the subject of review papers and entire chapters of books. This paper reviews some of the nice, useful properties of positive (semi)definite matrices, and insists in particular on (i) characterizations of positive (semi)definiteness and (ii) the geometrical properties of the set of positive semidefinite matrices. Some properties that turn out to be less well-known have here a special treatment. The use of these properties in optimization, as well as various references to applications, are spread all the way through. The "raison d'être" of this paper is essentially pedagogical; it adopts the viewpoint of variational analysis, shedding new light on the topic. Important, fruitful, and subtle, the positive semidefinite world is a good place to start with this domain of applied mathematics

Convergence of best entropy estimates

Abstract. Given a finite number of moments of an unknown density on a finite measure space, the best entropy estimate--that nonnegative density x with the given moments which minimizes the Boltzmann-Shannon entropy I(x): = x log x--is considered. A direct proof is given that I has the Kadec property in L--if Yn converges weakly to 37 and I(yn) converges to I(37), then yn converges to 37 in norm. As a corollary, it is obtained that, as the number of given moments increases, the best entropy estimates converge in LI norm to the best entropy estimate of the limiting problem, which is simply in the determined case. Furthermore, for classical moment problems on intervals with strictly positive and sufficiently smooth, error bounds and uniform convergence are actually obtained. Key words, moment problem, entropy, Kadec, partially finite program, normal convex integrand, duality AMS(MOS) subject classifications, primary 41A46, 05C38; secondary 08A45, 28A2

Strong rotundity and optimization

Standard techniques from the study of well-posedness show that if a fixed convex objective function is minimized in turn over a sequence of convex feasible regions converging Mosco to a limiting feasible region, then the optimal solutions converge in norm to the optimal solution of the limiting problem. Certain conditions on the objective function are needed as is a constraint qualification. If, as may easily occur in practice, the constraint qualification fails, stronger set convergence is required, together with stronger analytic/geometric properties of the objective function: strict convexity (to ensure uniqueness), weakly compact level sets (to ensure existence and weak convergence), and the Kadec property (to deduce norm convergence). By analogy with the L_p norms, such properties are termed "strong rotundity." A very simple characterization of strongly rotund integral functionals on L₁ is presented that shows, for example, that the Boltzmann-Shannon entropy ∫ x log x is strongly rotund. Examples are discussed, and the existence of everywhere- and densely-defined strongly rotund functions is investigated

Decomposition of multivariate functions

Given a bivariate function defined on some subset of the Cartesian product of two sets, it is natural to ask when that function can be decomposed as the sum of two univariate functions. In particular, is a pointwise limit of such functions itself decomposable? At first glance this might seem obviously true but, as we show, the possibilities are quite subtle. We consider the question of existence and uniqueness of such decompositions for this case and for many generalizations to multivariate functions and to cases where the sets and functions have topological or measure theoretic structure

North-Holland Partially finite convex programming, Part II" Explicit lattice models

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 of quasi 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 a 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

Partially finite convex programming, part I: quasi relative interiors and duality theory

We study convex programs that involve the minimization of a convex function over a convex subset of a topological vector space, subject to a finite number of linear inequalities. We develop the notion of the quasi relative interior of a convex set, an extension of the relative interior in finite dimensions. We use this idea in a constraint qualification for a fundamental Fenchel duality result, and then deduce duality results for these problems despite the almost invariable failure of the standard Slater condition. Part II of this work studies applications to more concrete models, whose dual problems are often finite-dimensional and computationally tractable

Partially-finite programming in L₁ and the existence of maximum entropy estimates

Best entropy estimation is a technique that has been widely applied in many areas of science. It consists of estimating an unknown density from some of its moments by maximizing some measure of the entropy of the estimate. This problem can be modelled as a partially-finite convex program, with an integrable function as the variable. A complete duality and existence theory is developed for this problem and for an associated extended problem which allows singular, measure-theoretic solutions. This theory explains the appearance of singular components observed in the literature when the Burg entropy is used. It also provides a unified treatment of existence conditions when the Burg, Boltzmann-Shannon, or some other entropy is used as the objective. Some examples are discussed

On the convergence of moment problems

We study the problem of estimating a nonnegative density, given a finite number of moments. Such problems arise in numerous practical applications. As the number of moments increases, the estimates will always converge weak * as measures, but need not converge weakly in L₁. This is related to the existence of functions on a compact metric space which are not essentially Riemann integrable (in some suitable sense). We characterize the type of weak convergence we can expect in terms of Riemann integrability, and in some cases give error bounds. When the estimates are chosen to minimize an objective function with weakly compact level sets (such as the Bolzmann-Shannon entropy) they will converge weakly in L₁. When an L_p norm (1 < p < ∞) is used as the objective, the estimates actually converge in norm. These results provide theoretical support to the growing popularity of such methods in practice