Convex envelopes for edge-concave functions

Deterministic global optimization algorithms frequently rely on the convex underestimation of nonconvex functions. In this paper we describe the structure of the polyhedral convex envelopes of edge-concave functions over polyhedral domains using geometric arguments. An algorithm for computing the facets of the convex envelope over hyperrectangles in ³ is described. Sufficient conditions are described under which the convex envelope of a sum of edge-concave functions may be shown to be equivalent to the sum of the convex envelopes of these functions.Author to whom all correspondence should be addressed.     

Convex envelopes of products of convex and component-wise concave functions

In this paper, we consider functions of the form f(x,y)=f(x)g(y){\phi(x,y)=f(x)g(y)} over a box, where f(x), x ? \mathbb R{f(x), x\in {\mathbb R}} is a nonnegative monotone convex function with a power or an exponential form, and g(y), y ? \mathbb Rⁿ{g(y), y\in {\mathbb R}^n} is a component-wise concave function which changes sign over the vertices of its domain. We derive closed-form expressions for convex envelopes of various functions in this category. We demonstrate via numerical examples that the proposed envelopes are significantly tighter than popular factorable programming relaxations.     

A simplified conjugation scheme for lower semi-continuous functions

We present two generalized conjugation schemes for lower semi-continuous functions defined on a real Banach space whose norm is Fréchet differentiable off the origin, and sketch their applications to optimization duality theory. Both approaches are based upon a new characterization of lower semi-continuous functions as pointwise suprema of a special class of continuous functions.     

Convex envelopes of bivariate functions through the solution of KKT systems

In this paper we exploit a slight variant of a result previously proved in Locatelli and Schoen (Math Program 144:65–91, 2014) to define a procedure which delivers the convex envelope of some bivariate functions over polytopes. The procedure is based on the solution of a KKT system and simplifies the derivation of the convex envelope with respect to previously proposed techniques. The procedure is applied to derive the convex envelope of the bilinear function xy over any polytope, and the convex envelope of functions \(x^n y^m\) over boxes.     

Convex envelopes of separable functions over regions defined by separable functions of the same type

In this paper we derive the convex envelope of separable functions obtained as a linear combination of strictly convex coercive one-dimensional functions over compact regions defined by linear combinations of the same one-dimensional functions. As a corollary of the main result, we are able to derive the convex envelope of any quadratic function (not necessarily separable) over any ellipsoid, and the convex envelope of some quadratic functions over a convex region defined by two quadratic constraints.     

Integral inclusions of upper semi-continuous or lower semi-continuous type

Topological results for set valued maps are used to establish existence results for integral inclusions of Volterra or Hammerstein type.

     

Convex functions and subharmonic functions

It is a classical result that a composition of a convex, increasing function and of a subharmonic function is subharmonic. We give related results for a composition of a convex function of several variables and of several subharmonic functions, thus imporving some recent results in this area.     

Convergence of sequences of semi-continuous functions

In this paper we investigate how three well-known modes of convergence for (real-valued) functions are related to one another. In particular, we consider order convergence, pointwise convergence and continuous convergence of sequences of nearly finite normal lower semi-continuous functions. There is a natural comparison to be made between the results we obtain for convergence of sequences of semi-continuous functions, and classic results on the convergence of sequences of measurable functions.     

Subdifferential set of the supremum of lower semi-continuous convex functions and the conical hull intersection property

In this paper we give some characterizations for the subdifferential set of the supremum of an arbitrary (possibly infinite) family of proper lower semi-continuous convex functions. This is achieved by means of formulae depending exclusively on the (exact) subdifferential sets and the normal cones to the domains of the involved functions. Our approach makes use of the concept of conical hull intersection property (CHIP, for short). It allows us to establish sufficient conditions guarantying explicit representations for this subdifferential set at any point of the effective domain of the supremum function. Research supported by grant SB2003-0344 of SEUI (MEC), Spain.     

Insertion of semi-continuous functions and sequences of sets

We investigate the relations between decreasing sequences of sets and the insertion of semi-continuous functions, and give some characterizations of countably metacompact spaces, countably paracompact spaces, monotonically countably paracompact spaces (MCP), monotonically countably metacompact spaces (MCM), perfectly normal spaces and stratifiable spaces.     

Additive envelopes of continuous functions

We present an iterative method for constructing additive envelopes of continuous functions on a compact set, with contact at a specified point. For elements of a class of submodular functions we provide closed-form expressions for such additive envelopes.     

Semi-stratifiable spaces and the insertion of semi-continuous functions

In this paper, we investigate the relations between the stratifiable structure of spaces and the insertion of semi-continuous functions and give some characterizations of perfect spaces, semi-stratifiable spaces and K-semi-stratifiable spaces.     

Convex spectral functions

In this paper we characterize all convex functionals defined on certain convex sets of hermitian matrices and which depend only on the eigenvalues of matrices. We extend these results to certain classes of non-negative matrices. This is done by formulating some new characterizations for the spectral radius of non-negative matrices, which are of independent interest.     

Fixed point theory for compact upper semi-continuous or lower semi-continuous set valued maps

Fixed point theory is presented for compact u.s.c. and l.s.c. set valued maps.

     

Convex extensions of the convex set valued maps

In this article the existence of the convex extension of convex set valued map is considered. Conditions are obtained, based on the notion of the derivative of set valued maps, which guarantee the existence of convex extension. The conditions are given, when the convex set valued map has no convex extension. The convex set valued map is specified, which is the maximal convex extension of the given convex set valued map and includes all other convex extensions. The connection between Lipschitz continuity and existence of convex extension of the given convex set valued map is studied.     

Convex envelopes generated from finitely many compact convex sets

We consider the problem of constructing the convex envelope of a lower semi-continuous function defined over a compact convex set. We formulate the envelope representation problem as a convex optimization problem for functions whose generating sets consist of finitely many compact convex sets. In particular, we consider nonnegative functions that are products of convex and component-wise concave functions and derive closed-form expressions for the convex envelopes of a wide class of such functions. Several examples demonstrate that these envelopes reduce significantly the relaxation gaps of widely used factorable relaxation techniques.