A geometrical analysis of the efficient outcome set in multiple objective convex programs with linear criterion functions

This article performs a geometrical analysis of the efficient outcome setY _E of a multiple objective convex program (MLC) with linear criterion functions. The analysis elucidates the facial structure ofY _E and of its pre-image, the efficient decision setX _E . The results show thatY _E often has a significantly-simpler structure thanX _E . For instance, although both sets are generally nonconvex and their maximal efficient faces are always in one-to-one correspondence, large numbers of extreme points and faces inX _E can map into non-facial subsets of faces inY _E , but not vice versa. Simple tests for the efficiency of faces in the decision and outcome sets are derived, and certain types of faces in the decision set are studied that are immune to a common phenomenon called collapsing. The results seem to indicate that significant computational benefits may potentially be derived if algorithms for problem (MLC) were to work directly with the outcome set of the problem to find points and faces ofY _E , rather than with the decision set.     

Vector optimization problems with quasiconvex constraints

Let X be a real linear space, a convex set, Y and Z topological real linear spaces. The constrained optimization problem min _C f(x), is considered, where f : X ₀→ Y and g : X ₀→ Z are given (nonsmooth) functions, and and are closed convex cones. The weakly efficient solutions (w-minimizers) of this problem are investigated. When g obeys quasiconvex properties, first-order necessary and first-order sufficient optimality conditions in terms of Dini directional derivatives are obtained. In the special case of problems with pseudoconvex data it is shown that these conditions characterize the global w-minimizers and generalize known results from convex vector programming. The obtained results are applied to the special case of problems with finite dimensional image spaces and ordering cones the positive orthants, in particular to scalar problems with quasiconvex constraints. It is shown, that the quasiconvexity of the constraints allows to formulate the optimality conditions using the more simple single valued Dini derivatives instead of the set valued ones.      

On the connectedness of the efficient set for strictly quasiconvex vector minimization problems

In this paper, we investigate the connectedness of the efficient solution set for vector minimization problems defined by a continuous vector-valued strictly quasiconvex functionf=(f ₁,...,f _m ) _T and a convex compact setX. It is shown that the efficient solution set is connected if one component off is strongly quasiconvex onX.The author would like to thank Professor H. P. Benson and the referees for many valuable comments and for pointing out some errors in the previous draft.Formerly, Assistant, Department of Applied Mathematics, Shanghai Jiao Tong University, Shanghai, China.     

Cross-cuts in the power set of an infinite set

In the power setP(E) of a setE, the sets of a fixed finite cardinalityk form across-cut, that is, a maximal unordered setC such that ifX, Y E satisfyXY, X someX inC, andY someY inC, thenXZY for someZ inC. ForE=, ₁, and ₂, it is shown with the aid of the continuum hypothesis thatP(E) has cross-cuts consisting of infinite sets with infinite complements, and somewhat stronger results are proved for and ₁.The work reported here has been partially supported by NSERC Grant No. A8054.     

Tiling convex sets by translates

LetK be a compact, convex subset ofE ^dwhich can be tiled by a finite number of disjoint (on interiors) translates of some compact setY. Then we may writeK=X+Y, whereX is finite. The possible structures forK, X andY are completely determined under these conditions.     

Topological direct sum decompositions of banach spaces

LetY andZ be two closed subspaces of a Banach spaceX such thatY≠lcub;0rcub; andY+Z=X. Then, ifZ is weakly countably determined, there exists a continuous projectionT inX such that ∥T∥=1,T(X)⊃Y, T ⁻¹(0)⊂Z and densT(X)=densY. It follows that every Banach spaceX is the topological direct sum of two subspacesX ₁ andX ₂ such thatX ₁ is reflexive and densX ₂**=densX**/X.     

The Measure of Noncompactness of Linear Operators between Spaces of Strongly C 1 Summable and Bounded Sequences

We give necessary and sufficient conditions for infinite matrices to map a sequence space X into a sequence space Y where X = l ₁ and Y = w ^p , w ^p , w ₀ ^p (1 p < ), or X = w ₀, w, w and Y = l _p (1 p ), or X = w ₀, w, w and Y = w ₀ ^p , w ^p , w ^p (1 p < ). Furthermore the Hausdorff measure of noncompactness is applied to give necessary and sufficient conditions for a linear operator between these spaces to be compact.     

Some results about nondominated solutions

We will prove that, in the case where all the domination cones are closed and convex, have a nonempty interior, and are totally ordered with respect to inclusion, the set of weakly nondominated outcomes of a nonempty convex compact setY R ⁿ is connected. Further, we give a relationship between weakly nondominated and nondominated solutions.The author would like to thank the referees for their valuable comments.     

On the martingale problem for Banach space valued stochastic differential equations

LetE denote a real separable Banach space and letZ=(Z(t, f) be a family of centered, homogeneous, Gaussian independent increment processes with values inE, indexed by timet0 and the continuous functionsf:[0,t] E. If the dependence ont andf fulfills some additional properties,Z is called a gaussian random field. For continuous, adaptedE-valued processesX a stochastic integral processY = ₀ ^. Z(t, X)(dt) is defined, which is a continuous local martingale with tensor quadratic variation[Y] = ₀ ^. Q(t, X)dt, whereQ(t, f) denotes the covariance operator ofZ(t, f).Y is called a solution of the homogeneous Gaussian martingale problem, ifY = ₀ ^. Z(t, Y)(dt). Such solutions occur naturally in connection with stochastic differential equations of the type (D):dX(t)=G(t, X) dt+Z(t, X)(dt), whereG is anE-valued vector field. It is shown that a solution of (D) can be obtained by a kind of variation of parameter method, first solving a deterministic integral equation only involvingG and then solving an associated homogeneous martingale problem.     

Contractibility of the Efficient Set in Strictly Quasiconcave Vector Maximization

In this paper, we investigate the contractibility of the efficient frontier in a vector maximization problem defined by a continuous vector-valued strictly quasiconcave function and a convex compact set D in ^p . It is shown that the efficient frontier is contractible if one of the components of g is strongly quasiconcave on X. This work extends a result by Sun (see Ref. 1), which confirms the connectedness of the efficient frontier.     

Isometric factorization of weakly compact operators and the approximation property

Using an isometric version of the Davis, Figiel, Johnson, and Pe?czyński factorization of weakly compact operators, we prove that a Banach spaceX has the approximation property if and only if, for every Banach spaceY, the finite rank operators of norm ≤1 are dense in the unit ball ofW(Y,X), the space of weakly compact operators fromY toX, in the strong operator topology. We also show that, for every finite dimensional subspaceF ofW(Y,X), there are a reflexive spaceZ, a norm one operatorJ:Y→Z, and an isometry Φ :F →W(Y,X) which preserves finite rank and compact operators so thatT=Φ(T) oJ for allT∈F. This enables us to prove thatX has the approximation property if and only if the finite rank operators form an ideal inW(Y,X) for all Banach spacesY.     

On the Density of Positive Proper Efficient Points in a Normed Space

In the context of vector optimization and generalizing cones with bounded bases, we introduce and study quasi-Bishop-Phelps cones in a normed space X. A dual concept is also presented for the dual space X*. Given a convex subset A of a normed space X partially ordered by a closed convex cone S with a base, we show that, if A is weakly compact, then positive proper efficient points are sequentially weak dense in the set E(A, S) of efficient points of A; in particular, the connotation weak dense in the above can be replaced by the connotation norm dense if S is a quasi-Bishop-Phelps cone. Dually, for a convex subset of X* partially ordered by the dual cone S ⁺, we establish some density results of positive weak* efficient elements of A in E(A, S ⁺).     

The Choquet integral representation in the affine vector-valued case

LetX be any compact convex subset of a locally convex Hausdorff space andE be a complex Banach space. We denote byA(X, E) the space of all continuous and affineE-valued functions defined onX. In this paper we prove thatX is a Choquet simplex if and only if the dual ofA(X, E) is isometrically isomorphic by a selection map toM _m (X, E*), the space ofE*-valued,w*-regular boundary measures onX. This extends and strengthens a result of G. M. Ustinov. To do this we show that for any compact convex setX, each element of the dual ofA(X, E) can be represented by a measure inM _m (X, E*) with the same norm, and this representation is unique if and only ifX is a Choquet simplex. We also prove that ifX is metrizable andE is separable then there exists a selection map from the unit ball of the dual ofA(X, E) into the unit ball ofM _m (X, E*) which is weak* to weak*-Borel measurable.This work will constitute a portion of the author's Ph.D. Thesis at the University of Illinois.     

First-order optimality conditions in set-valued optimization

A a set-valued optimization problem min _C F(x), x ∈X ₀, is considered, where X ₀ ⊂ X, X and Y are normed spaces, F: X ₀ ⊂ Y is a set-valued function and C ⊂ Y is a closed cone. The solutions of the set-valued problem are defined as pairs (x ⁰,y ⁰), y ⁰ ∈F(x ⁰), and are called minimizers. The notions of w-minimizers (weakly efficient points), p-minimizers (properly efficient points) and i-minimizers (isolated minimizers) are introduced and characterized through the so called oriented distance. The relation between p-minimizers and i-minimizers under Lipschitz type conditions is investigated. The main purpose of the paper is to derive in terms of the Dini directional derivative first order necessary conditions and sufficient conditions a pair (x ⁰, y ⁰) to be a w-minimizer, and similarly to be a i-minimizer. The i-minimizers seem to be a new concept in set-valued optimization. For the case of w-minimizers some comparison with existing results is done.     

The Jordan–von Neumann constants and fixed points for multivalued nonexpansive mappings

The purpose of this paper is to study the existence of fixed points for nonexpansive multivalued mappings in a particular class of Banach spaces. Furthermore, we demonstrate a relationship between the weakly convergent sequence coefficient WCS(X) and the Jordan–von Neumann constant C_NJ(X) of a Banach space X. Using this fact, we prove that if C_NJ(X) is less than an appropriate positive number, then every multivalued nonexpansive mapping has a fixed point where E is a nonempty weakly compact convex subset of a Banach space X, and KC(E) is the class of all nonempty compact convex subsets of E.     

On coverings of plane convex sets by translates of strips

In the euclidean planeE ² letS ₁,S ₂, ... be a sequence of strips of widthsw ₁,w ₂, .... It is shown thatE ² can be covered by translates of the stripsS _i if w ₁ ^3/2 = . Further results concern conditions in order that a compact convex domain inE ² can be covered by translates ofS ₁,S ₂, ....This research was supported by National Science Foundation Research Grant MCS 76-06111.     

Interior points of convex hulls

If a setX ⊂E ⁿ has non-emptyk-dimensional interior, or if some point isk-dimensional surrounded, then the classic theorem of E. Steinitz may be extended. For example ifX ⊂E ⁿ has int _k X ≠ 0, (0 ≦k≦n) and ifp ɛ int conX, thenp ɛ int conY for someY ⊂X with cardY≦2n−k+1.     

Approximation of the efficient point set by perturbation of the ordering cone

A vector optimization problem is given by a feasible setZ ⁿ , a vector-valued objective functionf: ^{n l} , and an ordering coneC ^l . We perturb the ordering cone in such a way that the weakly efficient points of the perturbed vector optimization problem given byZ, f, and the perturbed cone are efficient points of the original problem. Especially this means that scalarization methods, which compute in general only weakly efficient points, determine efficient points of the original problem, when they were applied to the perturbed problem.It turns out that the efficient points are the limits of weakly efficient points of the perturbed problems, letting the perturbation tend to zero. On the basis of this, a reference point algorithm is formulated. Finally, we apply this algorithm to a structural optimization problem.     

Strong duality for infinite-dimensional vector-valued programming problems

LetX,Y andZ be locally convex real topological vector spaces,A?X a convex subset, and letC?Y,E?Z be cones. Letf:X→Z beE-concave andg:X→Y beC-concave functions. We consider a concave programming problem with respect to an abstract cone and its strong dual problem as follows: $$\begin{gathered} (P)maximize f(x), subject to x \in A, g(x) \in C, \hfill \\ (SD)minimize \left\{ {\mathop \cup \limits_{\varphi \in C^ + } \max \{ (f + \varphi \circ g)(A):E\} } \right\}, \hfill \\ \end{gathered} $$ , whereC ⁺ denotes the set of all nonnegative continuous linear operators fromY toZ and (SD) is the strong dual problem to (P). In this paper, the authors find a necessary condition of strong saddle point for Problem (P) and establish the strong duality relationships between Problems (P) and (SD).     

A symmetric numerical range for matrices

Summary For each normv on ⁿ, we define a numerical rangeZ _v, which is symmetric in the sense thatZ _v=Z_vD, wherev ^D is the dual norm.We prove that, fora ⁿⁿ,Z _v(a) contains the classical field of valuesV(a). In the special case thatv is thel ₁-norm,Z _v(a) is contained in a setG(a) of Gershgorin type defined by C. R. Johnson.Whena is in the complex linear span of both the Hermitians and thev-Hermitians, thenZ _v(a),V(a) and the convex hull of the usualv-numerical rangeV _v(a) all coincide. We prove some results concerning points ofV(a) which are extreme points ofZ _v(a).Part of this research was done while the authors were at the Mathematische Institut, Technische Universität, München, West Germany. The first author presented these results at the Seminar on Matrix Theory (Positivity and Norms) held in Munich in December, 1974. The second author also acknowledges support from the National Science Foundation under grant GP 37978X.