Analyzing linear systems containing strict inequalities via evenly convex hulls

The evenly convex hull of a given set is the intersection of all the open halfspaces which contain such set (hence the convex hull is contained in the evenly convex hull). This paper deals with finite dimensional linear systems containing strict inequalities and (possibly) weak inequalities as well as equalities. The number of inequalities and equalities in these systems is arbitrary (possibly infinite). For such kind of systems a consistency theorem is provided and those strict inequalities (weak inequalities, equalities) which are satisfied for every solution of a given system are characterized. Such results are formulated in terms of the evenly convex hull of certain sets which depend on the coefficients of the system.     

Constraint Qualifications in Nonsmooth Multiobjective Optimization

For an inequality constrained nonsmooth multiobjective optimization problem involving locally Lipschitz functions, stronger KT-type necessary conditions and KT necessary conditions (which in the continuously differentiable case reduce respectively to the stronger KT conditions studied recently by Maeda and the usual KT conditions) are derived for efficiency and weak efficiency under several constraint qualifications. Stimulated by the stronger KT-type conditions, the notion of core of the convex hull of the union of finitely many convex sets is introduced. As main tool in the derivation of the necessary conditions, a theorem of the alternatives and a core separation theorem are also developed which are respectively extensions of the Motzkin transposition theorem and the Tucker theorem.     

Motzkin’s theorem of the alternative: a continuous-time generalization

We revisit Zalmai’s theorem, which is a partial generalization of Motzkin’s theorem of the alternative in the continuous-time setting. In particular, we provide two simple examples demonstrating that its existing proof is incorrect, and we demonstrate that a suitably modified variant of Zalmai’s theorem, concerned with the inconsistency of systems of convex inequalities and affine equalities, can be verified. We also derive two generalized variants of Motzkin’s theorem of the alternative in the continuous-time setting.     

Dual Characterizations of Set Containments with Strict Convex Inequalities

Characterizations of the containment of a convex set either in an arbitrary convex set or in the complement of a finite union of convex sets (i.e., the set, described by reverse-convex inequalities) are given. These characterizations provide ways of verifying the containments either by comparing their corresponding dual cones or by checking the consistency of suitable associated systems. The convex sets considered in this paper are the solution sets of an arbitrary number of convex inequalities, which can be either weak or strict inequalities. Particular cases of dual characterizations of set containments have played key roles in solving large scale knowledge-based data classification problems where they are used to describe the containments as inequality constraints in optimization problems. The idea of evenly convex set (intersection of open half spaces), which was introduced by W. Fenchel in 1952, is used to derive the dual conditions, characterizing the set containments.     

Dual characterizations of the set containments with strict cone-convex inequalities in Banach spaces

We give characterizations of the containment of a convex set either in an arbitrary convex set or in a set described by reverse cone-convex inequalities in Banach spaces. The convex sets under consideration are the solution sets of an arbitrary number of cone-convex inequalities, which can be either weak or strict inequalities. These characterizations provide ways of verifying the containments either by comparing their corresponding dual cones or by checking the consistency of suitable associated systems. Particular cases of dual characterizations of set containments have played key roles in solving large scale knowledge-based data classification problems, where they are used to describe the containments as inequality constraints in optimization problems. The concept of evenly convex set is used to derive the dual conditions, characterizing the set containments.      

On the Euler characteristic of finite unions of convex sets

The Euler characteristic plays an important role in many subjects of discrete and continuous mathematics. For noncompact spaces, its homological definition, being a homotopy invariant, seems not as important as its role for compact spaces. However, its combinatorial definition, as a finitely additive measure, proves to be more applicable in the study of singular spaces such as semialgebraic sets, finitely subanalytic sets, etc. We introduce an interesting integral by means of which the combinatorial Euler characteristic can be defined without the necessity of decomposition and extension as in the traditional treatment for polyhedra and finite unions of compact convex sets. Since finite unions of closed convex sets cannot be obtained by cutting convex sets as in the polyhedral case, a separate treatment of the Euler characteristic for functions generated by indicator functions of closed convex sets and relatively open convex sets is necessary, and this forms the content of this paper.     

Lipschitzian stability of parametric variational inequalities over generalized polyhedra in Banach spaces

This paper concerns the study of solution maps to parameterized variational inequalities over generalized polyhedra in reflexive Banach spaces. It has been recognized that generalized polyhedral sets are significantly different from the usual convex polyhedra in infinite dimensions and play an important role in various applications to optimization, particularly to generalized linear programming. Our main goal is to fully characterize robust Lipschitzian stability of the aforementioned solution maps entirely via their initial data. This is done on the basis of the coderivative criterion in variational analysis via efficient calculations of the coderivative and related objects for the systems under consideration. The case of generalized polyhedra is essentially more involved in comparison with usual convex polyhedral sets and requires developing elaborated techniques and new proofs of variational analysis.     

On angle sums and steiner points of polyhedra

In the first part, the Euler-Gram formula for the angle sum of a convex polytope is extended to cellular decompositions of arbitrary polyhedra. The second part contains an attempt to define Steiner points for unions of finitely many convex compact sets, and states some of their properties. Research supported by a fellowship of the Swiss National Foundation.     

Motzkin decomposition of closed convex sets

Theodore Motzkin proved, in 1936, that any polyhedral convex set can be expressed as the (Minkowski) sum of a polytope and a polyhedral convex cone. This paper provides five characterizations of the larger class of closed convex sets in finite dimensional Euclidean spaces which are the sum of a compact convex set with a closed convex cone. These characterizations involve different types of representations of closed convex sets as the support functions, dual cones and linear systems whose relationships are also analyzed in the paper. The obtaining of information about a given closed convex set F and the parametric linear optimization problem with feasible set F from each of its different representations, including the Motzkin decomposition, is also discussed.     

Strong conical hull intersection property, bounded linear regularity, Jameson’s property (G), and error bounds in convex optimization

The strong conical hull intersection property and bounded linear regularity are properties of a collection of finitely many closed convex intersecting sets in Euclidean space. These fundamental notions occur in various branches of convex optimization (constrained approximation, convex feasibility problems, linear inequalities, for instance). It is shown that the standard constraint qualification from convex analysis implies bounded linear regularity, which in turn yields the strong conical hull intersection property. Jameson’s duality for two cones, which relates bounded linear regularity to property (G), is re-derived and refined. For polyhedral cones, a statement dual to Hoffman’s error bound result is obtained. A sharpening of a result on error bounds for convex inequalities by Auslender and Crouzeix is presented. Finally, for two subspaces, property (G) is quantified by the angle between the subspaces. Received October 1, 1997 / Revised version received July 21, 1998? Published online June 11, 1999     

A generalization of the Shapley–Ichiishi result

The Shapley–Ichiishi result states that a game is convex if and only if the convex hull of marginal vectors equals the core. In this paper, we generalize this result by distinguishing equivalence classes of balanced games that share the same core structure. We then associate a system of linear inequalities with each equivalence class, and we show that the system defines the class. Application of this general theorem to the class of convex games yields an alternative proof of the Shapley–Ichiishi result. Other applications range from computation of stable sets in non-cooperative game theory to determination of classes of TU games on which the core correspondence is additive (even linear). For the case of convex games we prove that the theorem provides the minimal defining system of linear inequalities. An example shows that this is not necessarily true for other equivalence classes of balanced games.     

Deterministic global optimization with partition sets whose feasibility is not known: Application to concave minimization,reverse convex constraints,DC-programming,and Lipschitzian optimization

A crucial problem for many global optimization methods is how to handle partition sets whose feasibility is not known. This problem is solved for broad classes of feasible sets including convex sets, sets defined by finitely many convex and reverse convex constraints, and sets defined by Lipschitzian inequalities. Moreover, a fairly general theory of bounding is presented and applied to concave objective functions, to functions representable as differences of two convex functions, and to Lipschitzian functions. The resulting algorithms allow one to solve any global optimization problem whose objective function is of one of these forms and whose feasible set belongs to one of the above classes. In this way, several new fields of optimization are opened to the application of global methods.     

Nonnegative solvability of linear equations in certain ordered rings

In the integers and in certain densely ordered rings that are not fields, projections of the solution set of finitely many homogeneous weak linear inequalities may be defined by finitely many congruence inequalities, where a congruence inequality combines a weak inequality with a system of congruences. These results extend well-known facts about systems of weak linear inequalities over ordered fields and imply corresponding analogues of Farkas' Lemma on nonnegative solvability of systems of linear equations.

     

A characterization of vNM-stable sets for linear production games

We discuss linear production games or market games with a continuum of players which are represented as minima of finitely many nonatomic measures.?Within this context we consider vNM-Stable Sets according to von Neumann and Morgenstern. We classify or characterize all solutions of this type which are convex polyhedra, i.e., which are the convex hull of finitely many imputations. Specifically, in each convex polyhedral vNM-Stable Set (and not only in the symmetric ones), the different types of traders must organize themselves into cartels. The vNM-Stable Set is then the convex hull of the utility distributions of the cartels.?Using the results from the continuum, we obtain a similar characterization also for finite glove market games. Received December 1998/Revised version June 1999     

Primal and Dual Stability Results for Variational Inequalities

The purpose of this paper is to study the continuous dependence of solutions of variational inequalities with respect to perturbations of the data that are maximal monotone operators and closed convex functions. The constraint sets are defined by a finite number of linear equalities and non linear convex inequalities. Primal and dual stability results are given, extending the classical ones for optimization problems.     

Analytical Linear Inequality Systems and Optimization

In many interesting semi-infinite programming problems, all the constraints are linear inequalities whose coefficients are analytical functions of a one-dimensional parameter. This paper shows that significant geometrical information on the feasible set of these problems can be obtained directly from the given coefficient functions. One of these geometrical properties gives rise to a general purification scheme for linear semi-infinite programs equipped with so-called analytical constraint systems. It is also shown that the solution sets of such kind of consistent systems form a transition class between polyhedral convex sets and closed convex sets in the Euclidean space of the unknowns.     

A majorization–minimization algorithm for split feasibility problems

The classical multi-set split feasibility problem seeks a point in the intersection of finitely many closed convex domain constraints, whose image under a linear mapping also lies in the intersection of finitely many closed convex range constraints. Split feasibility generalizes important inverse problems including convex feasibility, linear complementarity, and regression with constraint sets. When a feasible point does not exist, solution methods that proceed by minimizing a proximity function can be used to obtain optimal approximate solutions to the problem. We present an extension of the proximity function approach that generalizes the linear split feasibility problem to allow for non-linear mappings. Our algorithm is based on the principle of majorization–minimization, is amenable to quasi-Newton acceleration, and comes complete with convergence guarantees under mild assumptions. Furthermore, we show that the Euclidean norm appearing in the proximity function of the non-linear split feasibility problem can be replaced by arbitrary Bregman divergences. We explore several examples illustrating the merits of non-linear formulations over the linear case, with a focus on optimization for intensity-modulated radiation therapy.     

The structure of weak Pareto solution sets in piecewise linear multiobjective optimization in normed spaces

In general normed spaces,we consider a multiobjective piecewise linear optimization problem with the ordering cone being convex and having a nonempty interior.We establish that the weak Pareto optimal solution set of such a problem is the union of finitely many polyhedra and that this set is also arcwise connected under the cone convexity assumption of the objective function.Moreover,we provide necessary and suffcient conditions about the existence of weak(sharp) Pareto solutions.     

Unique lifting of integer variables in minimal inequalities

This paper contributes to the theory of cutting planes for mixed integer linear programs (MILPs). Minimal valid inequalities are well understood for a relaxation of an MILP in tableau form where all the nonbasic variables are continuous; they are derived using the gauge function of maximal lattice-free convex sets. In this paper we study lifting functions for the nonbasic integer variables starting from such minimal valid inequalities. We characterize precisely when the lifted coefficient is equal to the coefficient of the corresponding continuous variable in every minimal lifting (This result first appeared in the proceedings of IPCO 2010). The answer is a nonconvex region that can be obtained as a finite union of convex polyhedra. We then establish a necessary and sufficient condition for the uniqueness of the lifting function.     

Partial Convexity

We study OC-convexity, which is defined by the intersection of conic semispaces of partial convexity. We investigate an optimization problem for OC-convex sets and prove a Krein--Milman type theorem for OC-convexity. The relationship between OC-convex and functionally convex sets is studied. Topological and numerical aspects, as well as separability properties are described. An upper estimate for the Carathéodory number for OC-convexity is found. On the other hand, it happens that the Helly and the Radon number for OC-convexity are infinite. We prove that the OC-convex hull of any finite set of points is the union of finitely many polyhedra.