Construction of functions with uniform sublevel sets

In this paper, we study extended real-valued functions with uniform sublevel sets. The sublevel sets are defined by a linear shift of a set in a specified direction. We prove that the class of these functions coincides with the class of Gerstewitz functionals. In this way, we obtain a formula for the construction of such functions. The sublevel sets of Gerstewitz functionals are characterized and illustrated by examples. The results contain statements for translative functions, which are just the functions with uniform sublevel sets considered. The investigated functions are defined on an arbitrary real vector space without assuming any topology or convexity.     

Nonconvex separation theorems and some applications in vector optimization

Separation theorems for an arbitrary set and a not necessarily convex set in a linear topological space are proved and applied to vector optimization. Scalarization results for weakly efficient points and properly efficient points are deduced.     

On the functions with pseudoconvex sublevel sets and optimality conditions

A new class of generalized convex functions, called the functions with pseudoconvex sublevel sets, is defined. They include quasiconvex ones. A complete characterization of these functions is derived. Further, it is shown that a continuous function admits pseudoconvex sublevel sets if and only if it is quasiconvex. Optimality conditions for a minimum of the nonsmooth nonlinear programming problem with inequality, equality and a set constraints are obtained in terms of the lower Hadamard directional derivative. In particular sufficient conditions for a strict global minimum are given where the functions have pseudoconvex sublevel sets.     

Unbounded critical points for a class of lower semicontinuous functionals

For a general class of lower semicontinuous functionals, we prove existence and multiplicity of critical points, which turn out to be unbounded solutions to the associated Euler equation. We apply a nonsmooth critical point theory developed in [10], [12] and [13] and applied in [8], [9] and [20] to treat the case of continuous functionals.     

Separation Theorem Based on the Quasirelative Interior and Application to Duality Theory

We present a separation theorem in which the classic interior is replaced by the quasirelative interior. We apply this result to a constrained problem in the infinite-dimensional convex case, making use of a condition replacing the standard Slater condition, which in some cases can fail.     

Basic properties of scalarizmg functionals for multiobjective optimization

If a partial ordering or preordering induced by a cone D defines a multi objective optimization problem, then scalarizing functionals for this problem shall posses two basic properties. D – monotonicity and D _e approximation. Several ways of constructing functionals with these properties are discussed in the paper.     

A note on the use of direct sufficient conditions in optimal control problems

It is always possible to transform a nonautonomous optimal control problem into an autonomous one. However, the direct sufficient conditions may yield no information when applied to this autonomous problem, even though they do allow one to conclude sufficiency when applied to the original nonautonomous problem.This research was supported by the Air Force Office of Scientific Research, under Grant No. AFOSR-76-2923.     

Separation of sets and Wolfe duality

Lagrangian duality can be derived from separation in the Image Space, namely the space where the images of the objective and constraining functions of the given extremum problem run. By exploiting such a result, we analyse the relationships between Wolfe and Mond-Weir duality and prove their equivalence in the Image Space under suitable generalized convexity assumptions.      

Minimization of Gerstewitz functionals extending a scalarization by Pascoletti and Serafini

Scalarization in vector optimization is often closely connected to the minimization of Gerstewitz functionals. In this paper, the minimizer sets of Gerstewitz functionals are investigated. Conditions are given under which such a set is nonempty and compact. Interdependencies between solutions of problems with different parameters or with different feasible point sets are shown. Consequences for the parameter control in scalarization methods are proved. It is pointed out that the minimization of Gerstewitz functionals is equivalent to an optimization problem which generalizes the scalarization by Pascoletti and Serafini. The results contain statements about minimizers of certain Minkowski functionals and norms. Some existence results for solutions of vector optimization problems are derived.     

Direct-prediction quasi-Newton methods in Hilbert space with applications to control problems

In this paper, the Hilbert-space analogue of a result of Huang, that all the methods in the Huang class generate the same sequence of points when applied to a quadratic functional with exact linear searches, is established. The convergence of a class of direct prediction methods based on some work of Dixon is then proved, and these methods are then applied to some control problems. Their performance is found to be comparable with methods involving exact linear searches.     

Direct prediction methods in Hilbert space with applications to control problems

In this paper, the convergence of variable-metric methods without line searches (direct prediction methods) applied to quadratic functionals on a Hilbert space is established. The methods are then applied to certain control problems with both free endpoints and fixed endpoints. Computational results are reported and compared with earlier results. The methods discussed here are found to compare favorably with earlier methods involving line searches and with other direct prediction quasi-Newton methods.     

Games with vector payoffs

Two-person games are defined in which the payoffs are vectors. Necessary and sufficient conditions for optimal mixed strategies are developed, and examples are presented.     

Separating sets by lower and upper semicontinuous functions with a closed graph

In this paper we characterize the pairs (A^?, A⁺) of disjoint subsets of perfectly normal topological space which can be separated by a lower and an upper semicontinuous function with a closed graph.     

A minimax theorem for functions with possibly nonconnected intersections of sublevel sets

We apply the selection theorem for multivalued mappings with paraconvex values (rather than various versions of KKM-principle) to prove several minimax theorems. In contrast with well-known minimax theorems for coordinatewise semicontinuous functions, in our theorems finite intersections of sublevel or uplevel sets can be nonempty and nonconnected.     

Variable metric methods in Hilbert space with applications to control problems

This paper is concerned with some of the most powerful methods of minimizing functionals on Hilbert space. It is established that certain classes of these methods are equivalent and their convergence is proved for certain nonquadratic functionals on a Hilbert space. A computational study of these methods applied to a control problem is also included with particular reference to the equivalence of methods mentioned above.     

Lower bounds and stochastic optimization algorithms for uniform designs with three or four levels

New lower bounds for three- and four-level designs under the centered -discrepancy are provided. We describe necessary conditions for the existence of a uniform design meeting these lower bounds. We consider several modifications of two stochastic optimization algorithms for the problem of finding uniform or close to uniform designs under the centered -discrepancy. Besides the threshold accepting algorithm, we introduce an algorithm named balance-pursuit heuristic. This algorithm uses some combinatorial properties of inner structures required for a uniform design. Using the best specifications of these algorithms we obtain many designs whose discrepancy is lower than those obtained in previous works, as well as many new low-discrepancy designs with fairly large scale. Moreover, some of these designs meet the lower bound, i.e., are uniform designs.

     

Maximal vectors and multi-objective optimization

Maximal vector andweak-maximal vector are the two basic notions underlying the various broader definitions (like efficiency, admissibility, vector maximum, noninferiority, Pareto's optimum, etc.) for optimal solutions of multi-objective optimization problems. Moreover, the understanding and characterization of maximal and weak-maximal vectors on the space of index vectors (vectors of values of the multiple objective functions) is fundamental and useful to the understanding and characterization of Pareto-optimal and weak-optimal solutions on the space of solutions.This paper is concerned with various characterizations of maximal and weak-maximal vectors in a general subset of the EuclideanN-space, and with necessary conditions for Pareto-optimal and weak-optimal solutions to a generalN-objective optimization problem having inequality, equality, and open-set constraints on then-space. A geometric method is described; the validity of scalarization by linear combination is studied, and weak conditioning by directional convexity is considered; local properties and a fundamental necessary condition are given. A necessary and sufficient condition for maximal vectors in a simplex or a polyhedral cone is derived. Necessary conditions for Pareto-optimal and weak-optimal solutions are given in terms of Lagrange multipliers, linearly independent gradients, Jacobian and Gramian matrices, and Jacobian determinants.Several advantages in approaching the multi-objective optimization problem in two steps (investigate optimal index vectors on the space of index vectors first, and study optimal solutions on the specific space of solutions next) are demonstrated in this paper.This work was supported by the National Science Foundation under Grant No. GK-32701.     

A result on sets,with applications to vector optimization

In this paper there is stated a result on sets in ordered linear spaces which can be used to show that some properties of the sets are inherited by their convex hulls under suitable conditions. As applications one gives a characterization of weakly efficient points and a duality result for nonconvex vector optimization problems.