General blow-ups of the projective plane

We study linear series on a projective plane blown up in a bunch of general points. Such series arise from plane curves of fixed degree with assigned fat base points. We give conditions (expressed as inequalities involving the number of points and the degree of the plane curves) on these series to be base point free, i.e. to define a morphism to a projective space. We also provide conditions for the morphism to be a higher order embedding. In the discussion of the optimality of obtained results we relate them to the Nagata Conjecture expressed in the language of Seshadri constants and we give a lower bound on these invariants.     

On optimality conditions in optimization problems on solutions of operator equations

We study optimization problems in the presence of connection in the form of operator equations defined in Banach spaces. We prove necessary conditions for optimality of first and second order, for example generalizing the Pontryagin maximal principle for these problems. It is not our purpose to state the most general necessary optimality conditions or to compile a list of all necessary conditions that characterize optimal control in any particular minimization problem. In the present article we describe schemes for obtaining necessary conditions for optimality on solutions of general operator equations defined in Banach spaces, and the scheme discussed here does not require that there be no global functional constraints on the controlling parameters. As an example, in a particular Banach space we prove an optimality condition using the Pontryagin-McShane variation. Bibliography: 20 titles. Translated fromProblemy Matematicheskoi Fiziki, 1998, pp. 55–67.     

Robust stochastic dominance and its application to risk-averse optimization

We introduce a new preference relation in the space of random variables, which we call robust stochastic dominance. We consider stochastic optimization problems where risk-aversion is expressed by a robust stochastic dominance constraint. These are composite semi-infinite optimization problems with constraints on compositions of measures of risk and utility functions. We develop necessary and sufficient conditions of optimality for such optimization problems in the convex case. In the nonconvex case, we derive necessary conditions of optimality under additional smoothness assumptions of some mappings involved in the problem.     

Optimality and duality in nonsmooth multiobjective fractional programming problem with constraints

In this paper, we establish some quotient calculus rules in terms of contingent derivatives for the two extended-real-valued functions defined on a Banach space and study a nonsmooth multiobjective fractional programming problem with set, generalized inequality and equality constraints. We define a new parametric problem associated with these problem and introduce some concepts for the (local) weak minimizers to such problems. Some primal and dual necessary optimality conditions in terms of contingent derivatives for the local weak minimizers are provided. Under suitable assumptions, sufficient optimality conditions for the local weak minimizers which are very close to necessary optimality conditions are obtained. An application of the result for establishing three parametric, Mond–Weir and Wolfe dual problems and several various duality theorems for the same is presented. Some examples are also given for our findings.

     

Sufficient Optimality Condition for Vector Optimization Problems under D.C. Data

In this paper, we establish sufficient optimality conditions for D.C. vector optimization problems. We also give an application to vector fractional mathematical programming in a ordred separable Hilbert space.     

Global Optimality Conditions for Some Classes of Optimization Problems

We establish new necessary and sufficient optimality conditions for global optimization problems. In particular, we establish tractable optimality conditions for the problems of minimizing a weakly convex or concave function subject to standard constraints, such as box constraints, binary constraints, and simplex constraints. We also derive some new necessary and sufficient optimality conditions for quadratic optimization. Our main theoretical tool for establishing these optimality conditions is abstract convexity.     

On Quantitative Stability in Optimization and Optimal Control

We study two continuity concepts for set-valued maps that play central roles in quantitative stability analysis of optimization problems: Aubin continuity and Lipschitzian localization. We show that various inverse function theorems involving these concepts can be deduced from a single general result on existence of solutions to an inclusion in metric spaces. As applications, we analyze the stability with respect to canonical perturbations of a mathematical program in a Hilbert space and an optimal control problem with inequality control constraints. For stationary points of these problems, Aubin continuity and Lipschitzian localization coincide; moreover, both properties are equivalent to surjectivity of the map of the gradients of the active constraints combined with a strong second-order sufficient optimality condition.     

Bilevel programming with discrete lower level problems

In this article, we investigate bilevel programming problems with discrete lower level and continuous upper level problems. We will analyse the structure of these problems and discuss both the optimistic and the pessimistic solution approach. Since neither the optimistic nor the pessimistic solution functions are in general lower semicontinuous, we introduce weak solution function. By using these functions we are able to discuss optimality conditions for local and global optimality.     

Second-order optimality conditions in set-valued optimization via asymptotic derivatives

In this article we give new second-order optimality conditions in set-valued optimization. We use the second-order asymptotic tangent cones to define second-order asymptotic derivatives and employ them to give the optimality conditions. We extend the well-known Dubovitskii–Milutin approach to set-valued optimization to express the optimality conditions given as an empty intersection of certain cones in the objective space. We also use some duality arguments to give new multiplier rules. By following the more commonly adopted direct approach, we also give optimality conditions in terms of a disjunction of certain cones in the image space. Several particular cases are discussed.     

Liberating the Subgradient Optimality Conditions from Constraint Qualifications

In convex optimization the significance of constraint qualifications is evidenced by the simple duality theory, and the elegant subgradient optimality conditions which completely characterize a minimizer. However, the constraint qualifications do not always hold even for finite dimensional optimization problems and frequently fail for infinite dimensional problems. In the present work we take a broader view of the subgradient optimality conditions by allowing them to depend on a sequence of ε-subgradients at a minimizer and then by letting them to hold in the limit. Liberating the optimality conditions in this way permits us to obtain a complete characterization of optimality without a constraint qualification. As an easy consequence of these results we obtain optimality conditions for conic convex optimization problems without a constraint qualification. We derive these conditions by applying a powerful combination of conjugate analysis and ε-subdifferential calculus. Numerical examples are discussed to illustrate the significance of the sequential conditions.     

A Unified Approach for Constrained Extremum Problems: Image Space Analysis

In this paper, by exploiting the image space analysis we investigate a class of constrained extremum problems, the constraining function of which is set-valued. We show that a (regular) linear separation in the image space is equivalent to the existence of saddle points of Lagrangian and generalized Lagrangian functions and we also give Lagrangian type optimality conditions for the class of constrained extremum problems under suitable generalized convexity and compactness assumptions. Moreover, we consider an exact penalty problem for the class of constrained extremum problems and prove that it is equivalent to the existence of a regular linear separation under suitable generalized convexity and compactness assumptions.     

Optimal control problems on manifolds: a dynamic programming approach

Dynamic programming identifies the value function of continuous time optimal control with a solution to the Hamilton-Jacobi equation, appropriately defined. This relationship in turn leads to sufficient conditions of global optimality, which have been widely used to confirm the optimality of putative minimisers. In continuous time optimal control, the dynamic programming methodology has been used for problems with state space a vector space. However there are many problems of interest in which it is necessary to regard the state space as a manifold. This paper extends dynamic programming to cover problems in which the state space is a general finite-dimensional C^∞ manifold. It shows that, also in a manifold setting, we can characterise the value function of a free time optimal control problem as a unique lower semicontinuous, lower bounded, generalised solution of the Hamilton-Jacobi equation. The application of these results is illustrated by the investigation of minimum time controllers for a rigid pendulum.     

Inhomogeneous convex approximations of nonsmooth functions

In this paper we introduce notions of inhomogeneous upper convex and lower concave approximations of an increment of a nonsmooth function defined on a normed space and study exhaustive families of these approximations. In terms of the introduced notions we establish optimality conditions for various constrained and unconstrained extremum problems.     

Abstract convex approximations of nonsmooth functions

In the article we use abstract convexity theory in order to unify and generalize many different concepts of nonsmooth analysis. We introduce the concepts of abstract codifferentiability, abstract quasidifferentiability and abstract convex (concave) approximations of a nonsmooth function mapping a topological vector space to an order complete topological vector lattice. We study basic properties of these notions, construct elaborate calculus of abstract codifferentiable functions and discuss continuity of abstract codifferential. We demonstrate that many classical concepts of nonsmooth analysis, such as subdifferentiability and quasidifferentiability, are particular cases of the concepts of abstract codifferentiability and abstract quasidifferentiability. We also show that abstract convex and abstract concave approximations are a very convenient tool for the study of nonsmooth extremum problems. We use these approximations in order to obtain various necessary optimality conditions for nonsmooth nonconvex optimization problems with the abstract codifferentiable or abstract quasidifferentiable objective function and constraints. Then, we demonstrate how these conditions can be transformed into simpler and more constructive conditions in some particular cases.     

Inverse stochastic dominance constraints and rank dependent expected utility theory

We consider optimization problems with second order stochastic dominance constraints formulated as a relation of Lorenz curves. We characterize the relation in terms of rank dependent utility functions, which generalize Yaari's utility functions. We develop optimality conditions and duality theory for problems with Lorenz dominance constraints. We prove that Lagrange multipliers associated with these constraints can be identified with rank dependent utility functions. The problem is numerically tractable in the case of discrete distributions with equally probable realizations. Research supported by the NSF awards DMS-0303545, DMS-0303728, DMI-0354500 and DMI-0354678.     

Stochastic multiobjective problems with complementarity constraints and applications in healthcare management

We consider a class of stochastic multiobjective problems with complementarity constraints (SMOPCCs) in this paper. We derive the first-order optimality conditions including the Clarke/Mordukhovich/strong-type stationarity in the Pareto sense for the SMOPCC. Since these first-order optimality systems involve some unknown index sets, we reformulate them as nonlinear equations with simple constraints. Then, we introduce an asymptotic method to solve these constrained equations. Furthermore, we apply this methodology results to a patient allocation problem in healthcare management.     

Fuzzy optimal control problem with non-linear functional

In this paper, we introduce a new space of fuzzy numbers equipped with a scalar product defined in this space. The notion of a derivative of a fuzzy function in this space is defined. By employing these notions, an optimal control problem with non-linear functional is formulated and an optimality condition is obtained in the form of maximum principle. Using this result, the numerical algorithm is offered for the solution of such problems.     

A dual dynamic programming for minimax optimal control problems governed by parabolic equation

In the article, minimax optimal control problems governed by parabolic equations are considered. We apply a new dual dynamic programming approach to derive sufficient optimality conditions for such problems. The idea is to move all the notions from a state space to a dual space and to obtain a new verification theorem providing the conditions, which should be satisfied by a solution of the dual partial differential equation of dynamic programming. We also give sufficient optimality conditions for the existence of an optimal dual feedback control and some approximation of the problem considered, which seems to be very useful from a practical point of view.     

Optimality conditions with state constraints for semilinear systems determined by operator valued measures

In this paper we develop the necessary conditions of optimality for a class of distributed parameter systems (partial differential equations) determined by operator valued measures and controlled by vector measures. Based on some recent results on existence of optimal controls from the space of vector measures, we develop necessary conditions of optimality for a class of control problems. The main results are the necessary conditions of optimality for problems without state constraints and those with state constraints. Also, a conceptual algorithm along with a brief discussion of its convergence is presented.     

On Optimization Problems with Set-Valued Objective Maps: Existence and Optimality

In this paper, we consider constrained optimization problems with set-valued objective maps. First, we define three types of quasi orderings on the set of all non-empty subsets of n-dimensional Euclidean space. Second, by using these quasi orderings, we define the concepts of lower semi-continuity for set-valued maps and investigate their properties. Finally, based on these results, we define the concepts of optimal solutions to constrained optimization problems with set-valued objective maps and we give some conditions under which these optimal solutions exist to the problems and give necessary and sufficient conditions for optimality.