Existence and sum decomposition of vertex polyhedral convex envelopes

Convex envelopes are a very useful tool in global optimization. However finding the exact convex envelope of a function is a difficult task in general. This task becomes considerably simpler in the case where the domain is a polyhedron and the convex envelope is vertex polyhedral, i.e., has a polyhedral epigraph whose vertices correspond to the vertices of the domain. A further simplification is possible when the convex envelope is sum decomposable, i.e., the convex envelope of a sum of functions coincides with the sum of the convex envelopes of the summands. In this paper we provide characterizations and sufficient conditions for the existence of a vertex polyhedral convex envelope. Our results extend and unify several results previously obtained for special cases of this problem. We then characterize sum decomposability of vertex polyhedral convex envelopes, and we show, among else, that the vertex polyhedral convex envelope of a sum of functions coincides with the sum of the vertex polyhedral convex envelopes of the summands if and only if the latter sum is vertex polyhedral.     

Semidefinite Relaxations of Fractional Programs via Novel Convexification Techniques

In a recent work, we introduced the concept of convex extensions for lower semi-continuous functions and studied their properties. In this work, we present new techniques for constructing convex and concave envelopes of nonlinear functions using the theory of convex extensions. In particular, we develop the convex envelope and concave envelope of z=x/y over a hypercube. We show that the convex envelope is strictly tighter than previously known convex underestimators of x/y. We then propose a new relaxation technique for fractional programs which includes the derived envelopes. The resulting relaxation is shown to be a semidefinite program. Finally, we derive the convex envelope for a class of functions of the type f(x,y) over a hypercube under the assumption that f is concave in x and convex in y.     

Convex Envelopes of Monomials of Odd Degree

Convex envelopes of nonconvex functions are widely used to calculate lower bounds to solutions of nonlinear programming problems (NLP), particularly within the context of spatial Branch-and-Bound methods for global optimization. This paper proposes a nonlinear continuous and differentiable convex envelope for monomial terms of odd degree, x ^2k+1, where k N and the range of x includes zero. We prove that this envelope is the tightest possible. We also derive a linear relaxation from the proposed envelope, and compare both the nonlinear and linear formulations with relaxations obtained using other approaches.     

Explicit convex and concave envelopes through polyhedral subdivisions

In this paper, we derive explicit characterizations of convex and concave envelopes of several nonlinear functions over various subsets of a hyper-rectangle. These envelopes are obtained by identifying polyhedral subdivisions of the hyper-rectangle over which the envelopes can be constructed easily. In particular, we use these techniques to derive, in closed-form, the concave envelopes of concave-extendable supermodular functions and the convex envelopes of disjunctive convex functions.     

Trilinear Monomials with Mixed Sign Domains: Facets of the Convex and Concave Envelopes

Convex underestimators of nonconvex functions, frequently used in deterministic global optimization algorithms, strongly influence their rate of convergence and computational efficiency. A good convex underestimator is as tight as possible and introduces a minimal number of new variables and constraints. Multilinear monomials over a coordinate aligned hyper-rectangular domain are known to have polyhedral convex envelopes which may be represented by a finite number of facet inducing inequalities. This paper describes explicit expressions defining the facets of the convex and concave envelopes of trilinear monomials over a box domain with bounds of opposite signs for at least one variable. It is shown that the previously used approximations based on the recursive use of the bilinear construction rarely yield the convex envelope itself.     

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.     

On Some Generalized Polyhedral Convex Constructions

Generalized polyhedral convex sets, generalized polyhedral convex functions on locally convex Hausdorff topological vector spaces, and the related constructions such as sum of sets, sum of functions, directional derivative, infimal convolution, normal cone, conjugate function, subdifferential are studied thoroughly in this paper. Among other things, we show how a generalized polyhedral convex set can be characterized through the finiteness of the number of its faces. In addition, it is proved that the infimal convolution of a generalized polyhedral convex function and a polyhedral convex function is a polyhedral convex function. The obtained results can be applied to scalar optimization problems described by generalized polyhedral convex sets and generalized polyhedral convex functions.     

On solving a D.C. programming problem by a sequence of linear programs

We are dealing with a numerical method for solving the problem of minimizing a difference of two convex functions (a d.c. function) over a closed convex set in ⁿ . This algorithm combines a new prismatic branch and bound technique with polyhedral outer approximation in such a way that only linear programming problems have to be solved.Parts of this research were accomplished while the third author was visiting the University of Trier, Germany, as a fellow of the Alexander von Humboldt foundation.     

Finite Order Rank-One Convex Envelopes and Computation of Microstructures with Laminates in Laminates

Finite order rank-one convex envelopes are introduced and it is shown that the i-th order laminated microstructures, or laminates in laminates, can be solved by any of the k-th order rank-one convex envelopes with k i. It is also shown that in finite element approximations of microstructures, replacing the non-quasiconvex potential energy density by its k-th order rank-one convex envelope, one can generally obtain sharper numerical results. Especially, for crystalline microstructures with laminates in laminates of order no greater than k + 1, numerical results with up to the computer precision can be obtained. Numerical examples on the first and second order rank-one convex envelopes for the Ericksen-James two-dimensional model for elastic crystals are given. A numerical example on finite element approximations of a crystalline microstructure by using the first order rank-one convex envelope and the periodic relaxation method is also presented. The methods turn out to be very successful for microstructures with laminates in laminates.     

An explicit characterization of the convex envelope of a bivariate bilinear function over special polytopes

In this paper, we constructively derive an explicit characterization of the convex envelope of a bilinear function over a special type of polytope in ². Our motivation stems from the use of such functions for deriving strengthened lower bounds within the context of a branch-and-bound algorithm for solving bilinear programming problems. For the case of polytopes with no edges having finite positive slopes, that is polytopes with downward sloping edges (which we call D-polytopes), we obtain a direct, explicit characterization of the convex envelope. This case subsumes the analysis of Al-Khayyal and Falk (1983) for constructing the convex envelope of a bilinear function over a rectangle in ². For non-D-polytopes, the analysis is more complex. We propose three strategies for this case based on (i) encasing the region in a D-polytope, (ii) employing a discretization technique, and (iii) providing an explicit characterization over a triangle along with a triangular decomposition approach. The analysis is illustrated using numerical examples.     

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.     

Computing a global optimal solution to a design centering problem

In this paper we present a method for solving a special three-dimensional design centering problem arising in diamond manufacturing: Find inside a given (not necessarily convex) polyhedral rough stone the largest diamond of prescribed shape and orientation. This problem can be formulated as the one of finding a global maximum of a difference of two convex functions over ³ and can be solved efficiently by using a global optimization algorithm provided that the objective function of the maximization problem can be easily evaluated. Here we prove that with the information available on the rough stone and on the reference diamond, evaluating the objective function at a pointx amounts to computing the distance, with respect to a Minkowski gauge, fromx to a finite number of planes. We propose a method for finding these planes and we report some numerical results.     

A Convex Envelope Formula for Multilinear Functions

Convex envelopes of multilinear functions on a unit hypercube arepolyhedral. This well-known fact makes the convex envelopeapproximation very useful in the linearization of non-linear 0–1programming problems and in global bilinear optimization. This paperpresents necessary and sufficient conditions for a convex envelope to be apolyhedral function and illustrates how these conditions may be used inconstructing of convex envelopes. The main result of the paper is a simpleanalytical formula, which defines some faces of the convex envelope of amultilinear function. This formula proves to be a generalization of the wellknown convex envelope formula for multilinear monomial functions.     

Some results on the strength of relaxations of multilinear functions

We study approaches for obtaining convex relaxations of global optimization problems containing multilinear functions. Specifically, we compare the concave and convex envelopes of these functions with the relaxations that are obtained with a standard relaxation approach, due to McCormick. The standard approach reformulates the problem to contain only bilinear terms and then relaxes each term independently. We show that for a multilinear function having a single product term, this approach yields the convex and concave envelopes if the bounds on all variables are symmetric around zero. We then review and extend some results on conditions when the concave envelope of a multilinear function can be written as a sum of concave envelopes of its individual terms. Finally, for bilinear functions we prove that the difference between the concave upper bounding and convex lower bounding functions obtained from the McCormick relaxation approach is always within a constant of the difference between the concave and convex envelopes. These results, along with numerical examples we provide, give insight into how to construct strong relaxations of multilinear functions.     

Convex extensions and envelopes of lower semi-continuous functions

 We define a convex extension of a lower semi-continuous function to be a convex function that is identical to the given function over a pre-specified subset of its domain. Convex extensions are not necessarily constructible or unique. We identify conditions under which a convex extension can be constructed. When multiple convex extensions exist, we characterize the tightest convex extension in a well-defined sense. Using the notion of a generating set, we establish conditions under which the tightest convex extension is the convex envelope. Then, we employ convex extensions to develop a constructive technique for deriving convex envelopes of nonlinear functions. Finally, using the theory of convex extensions we characterize the precise gaps exhibited by various underestimators of $x/y$ over a rectangle and prove that the extensions theory provides convex relaxations that are much tighter than the relaxation provided by the classical outer-linearization of bilinear terms. Received: December 2000 / Accepted: May 2002 Published online: September 5, 2002 RID="*" ID="*" The research was funded in part by a Computational Science and Engineering Fellowship to M.T., and NSF CAREER award (DMI 95-02722) and NSF/Lucent Technologies Industrial Ecology Fellowship (NSF award BES 98-73586) to N.V.S. Key words. convex hulls and envelopes – multilinear functions – disjunctive programming – global optimization     

Tight convex underestimators for {{\mathcal C}^2}-continuous problems: I. univariate functions

A novel method for the convex underestimation of univariate functions is presented in this paper. The method is based on a piecewise application of the well-known αBB underestimator, which produces an overall underestimator that is piecewise convex. Subsequently, two algorithms are used to identify the linear segments needed for the construction of its -continuous convex envelope, which is itself a valid convex underestimator of the original function. The resulting convex underestimators are very tight, and their tightness benefits from finer partitioning of the initial domain. It is theoretically proven that there is always some finite level of partitioning for which the method yields the convex envelope of the function of interest. The method was applied on a set of univariate test functions previously presented in the literature, and the results indicate that the method produces convex underestimators of high quality in terms of both lower bound and tightness over the whole domain under consideration.     

Polyedrische 2-Mannigfaltigkeiten mit wenigen nicht-konvexen Ecken

LetP denote a polyhedral 2-manifold in ³, i.e. a 2-dimensional cell-complex in ³ whose underlying point-set is a closed connected 2-manifold. A vertexv ofP is called convex if at least one of the two components into whichP divides a sufficiently small ball centered atv is convex. It is shown that every polyhedral 2-manifold in ³ of genusg>–1 contains at least five non-convex vertices and that for every positive integerg this bound is attained, i.e. there exists a polyhedral 2-manifold in ³ of genusg with precisely five non-convex vertices.

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Shelling pseudopolyhedra

The notion of shellability originated in the context of polyhedral complexes and combinatorial topology. An abstraction of this concept for graded posets (i.e., graded partially ordered sets) was recently introduced by Björner and Wachs first in the finite case [1] and then with Walker in the infinite case [11]. Many posets arising in combinatorics and in convex geometry were investigated and some proved to be shellable. A key achievement was the proof by Bruggesser and Mani that boundary complexes of convex polytopes are shellable [4].We extend here the result of Bruggesser and Mani to polyhedral complexes arising as boundary complexes of more general convex sets, called pseudopolyhedra, with suitable asymptotic behavior. This includes a previous result on tilings of a Euclidean space ^d which are projections of the boundary of a (d+1)-pseudopolyhedron [7].     

Analysis of Bounds for Multilinear Functions

We analyze four bounding schemes for multilinear functions and theoretically compare their tightness. We prove that one of the four schemes provides the convex envelope and that two schemes provide the concave envelope for the product of p variables over .