Separation Theorem For Convex Sets Research Articles

The nearest point problems in fuzzy quasi-normed spaces

<abstract> <p>Motivated by the fact that the fuzzy quasi-normed space provides a suitable framework for complexity analysis and has important roles in discussing some questions in theoretical computer science, this paper aims to study the nearest point problems in fuzzy quasi-normed spaces. First, by using the theory of dual space and the separation theorem of convex sets, the properties of the fuzzy distance from a point to a set in a fuzzy quasi-normed space are studied comprehensively. Second, more properties of the nearest point are given, and the existence, uniqueness, characterizations, and qualitative properties of the nearest points are obtained. The results obtained in this paper are of great significance for expanding the application fields of optimization theory.</p> </abstract>

Density and connectedness of optimal points with respect to improvement sets

Based on the improvement set E, this paper aims at investigating density and connectedness of the sets of E-optimal points, weak E-optimal points, E-quasi-optimal points, E-Benson proper optimal points, E-super optimal points and E-strictly optimal points. By virtue of the separation theorem for convex sets, we obtain the scalarizaiton results for the sets of E-optimal points, weak E-optimal points, E-Benson proper optimal points, E-super optimal points and E-strictly optimal points. Then we make a new attempt to establish some density theorems for the sets of these optimal points. Finally, we investigate connectedness and arcwise connectedness of the sets of various notions of E-optimality by using the scalarization method and the density theorems.

Connectedness of the approximate solution sets for set optimization problems

ABSTRACT In this paper, we deal with connectedness of the approximate solution sets for set optimization problems by using the scalarization method. We first obtain two scalarization results for the sets of l-minimal approximate solutions and weak l-minimal approximate solutions for set optimization problems without using the convexity of the objective mapping and the separation theorem of convex sets. Then, we establish connectedness of the sets of l-minimal approximate solutions and weak l-minimal approximate solutions for set optimization problems by utilizing the scalarization results. Finally, we derive arcwise connectedness of the set of weak l-minimal approximate solutions for set optimization problems.

Minimax Theorem for Two Set-Valued Mappings with Nonconvex Domains

In this paper, we obtain the operational properties of the set (MaxWA, and MinWA) of a set A in the sense of vector optimization and its’ applications to the judgment of the convexity and concavity of a set-valued mapping. By using the separation theorem of convex sets, some minimax theorems of two set-valued mappings have proved in the nonconvex sets. Simultaneously, two examples are given to show our results.

Convex MPC for exclusion constraints

This article develops model predictive control for exclusion constraints with a priori guaranteed strong system theoretic properties, which is implementable via computationally highly efficient, strictly convex quadratic programming. The proposed approach deploys safe tubes in order to ensure intrinsically nonconvex exclusion constraints via closed polyhedral constraints. A safe tube is constructed by utilizing the separation theorem for convex sets, and it is practically obtained from the solution of a strictly convex quadratic programming problem. A safe tube is deployed to efficiently optimize a predicted finite horizon control process via another strictly convex quadratic programming problem.

Optimality Conditions for Strictly Efficient Solutions in Set-valued Optimization

A new kind of tangent derivative, M-derivative, for set-valued function is introduced with help of a modified Dubovitskij-Miljutin cone. Several generalized pseudoconvex set-valued functions are introduced. When both the objective function and constraint function are M-derivative, under the assumption of near conesubconvexlikeness, by applying properties of the set of strictly efficient points and a separation theorem for convex sets, Fritz John and Kuhn-Tucker necessary optimality conditions are obtained for a point pair to be a strictly efficient element of set-valued optimization problem. Under the assumption of generalized pseudoconvexity, a Kuhn-Tucker sufficient optimality condition is obtained for a point pair to be a strictly efficient element of set-valued optimization problem.

Algebraic core and convex calculus without topology

ABSTRACT In this paper we study the concept of algebraic core for convex sets in general vector spaces without any topological structure and then present its applications to problems of convex analysis and optimization. Deriving the equivalence between the Hahn-Banach theorem and and a simple version of the separation theorem of convex sets in vector spaces allows us to develop a geometric approach to generalized differential calculus for convex sets, set-valued mappings, and extended-real-valued functions with qualification conditions formulated in terms of algebraic cores for such objects. We also obtain a precise formula for computing the subdifferential of optimal value functions associated with convex problems of parametric optimization in vector spaces. Functions of this type play a crucial role in many aspects of convex optimization and its applications.

Approximate Benson efficient solutions for set-valued equilibrium problems

In locally convex Hausdorff topological vector spaces, the approximate Benson efficient solution is proposed for set-valued equilibrium problems and its relationship to the Benson efficient solution is discussed. Under the assumption of generalized convexity, by using a separation theorem for convex sets, Kuhn–Tucker-type and Lagrange-type optimality conditions for set-valued equilibrium problems are established, respectively.

Characterization of ℓ1 minimizer in one-bit compressed sensing

The conception of one-bit compressive sensing (one-bit CS) was first introduced in 2008 by Boufounos and Baraniuk [1-Bit compressive sensing, in Proc. Conf. on Information Science and Systems (CISS, Princeton, NJ, 2008)]. Since then, many efficient algorithms have been developed for dealing with the one-bit CS problem. However, few theoretical results are available on one-bit CS. In this paper, we focus on one-bit CS theory with its relaxation model [Formula: see text] and present a necessary and sufficient condition such that the signal [Formula: see text] is the unique [Formula: see text] minimizer in noiseless one-bit CS (Theorem 3). Moreover, by using the improved separation theorem of convex sets (Theorem 6), we completely characterize the [Formula: see text] minimizer in one-bit CS (Theorem 2). Finally, as an application of Theorem 2, the [Formula: see text] minimizer for the considered model can, in general, be a non-sparse vector.

Second-Order M-Composed Tangent Derivative and Its Applications

A new kind of second-order tangent derivative, second-order [Formula: see text]-composed tangent derivative, for a set-valued function is introduced with help of a modified Dubovitskij–Miljutin cone. By using the concept, several generalized convex set-valued functions are introduced. When both the objective function and constrained function are second-order [Formula: see text]-composed derivable, under the assumption of nearly cone-subconvexlikeness, by applying a separation theorem for convex sets, Fritz John and Kuhn–Tucker second-order necessary optimality conditions are obtained for a point pair to be a weak minimizer of set-valued optimization problem. Under the assumption of generalized pseudoconvexity, a Kuhn–Tucker second-order sufficient optimality condition is obtained for a point pair to be a weak minimizer of set-valued optimization problem. A unified second-order necessary and sufficient optimality condition is derived in terms of second-order [Formula: see text]-composed tangent derivatives.

Approximate weakly efficient solutions of set-valued vector equilibrium problems

In this paper, we introduce a new kind of approximate weakly efficient solutions to the set-valued vector equilibrium problems with constraints in locally convex Hausdorff topological vector spaces; then we discuss a relationship between the weakly efficient solutions and approximate weakly efficient solutions. Under the assumption of near cone-subconvexlikeness, by using the separation theorem for convex sets we establish Kuhn–Tucker-type and Lagrange-type optimality conditions for set-valued vector equilibrium problems, respectively.

Second-Order Karush–Kuhn–Tucker Optimality Conditions for Set-Valued Optimization Subject to Mixed Constraints

In this paper, we propose the concept of second-order composed adjacent contingent derivatives for set-valued maps and hence discuss the included relationship to the second-order composed contingent derivatives. Under the appropriate conditions, Lipschitz properties of the first order derivatives and a chain rule for such second-order composed contingent derivatives are demonstrated. By virtue of second-order composed adjacent contingent derivatives and second-order composed contingent derivatives, we establish second-order Karush–Kuhn–Tucker sufficient and necessary optimality conditions for a set-valued optimization problem subject to mixed constraints. Applying a separation theorem for convex sets and cone-Aubin properties, we address stronger necessary optimality conditions and extend the second-order Kurcyusz–Robinson–Zowe regularity assumption for a problem with mixed constraints, in which the derivatives of the objective and the constraint functions are considered in separated ways. When the results regress to vector optimization, we also extend and improve some recent existing results.

A new kind of inner superefficient points

In this paper, some properties of the interior of positive dual cones are discussed. With the help of dilating cones, a new notion of inner superefficient points for a set is introduced. Under the assumption of near cone-subconvexlikeness, by applying the separation theorem for convex sets, the relationship between inner superefficient points and superefficient points is established. Compared to other approximate points in the literature, inner superefficient points in this paper are really ‘approximate’.

Sensitivity analysis in set-valued optimization under strictly minimal efficiency

In this paper, the behavior of the perturbation map is analyzed quantitatively by virtue of contingent derivatives and generalized contingent epiderivatives for the set-valued maps under strictly minimal efficiency. The purpose of this paper is to provide some well-known results concerning sensitivity analysis by applying a separation theorem for convex sets. When the results regress to multiobjective optimization, some related conclusions are obtained in a multiobjective programming problem.

New Second-Order Tangent Epiderivatives and Applications to Set-Valued Optimization

A new second-order tangent set is introduced, with which a new second-order tangent epiderivative is also introduced for a set-valued map. Applying a separation theorem for convex sets, second-order Fritz John and Kuhn---Tucker necessary optimality conditions are obtained for a point pair to be a weak minimizer of set-valued optimization problem. Under the assumption of lower semicontinuous, a second-order Kuhn---Tucker sufficient optimality condition is obtained for a point pair to be a weak minimizer of set-valued optimization problem.

Convexity and constructive infima

We show constructively that every quasi-convex uniformly continuous function \(f : \mathrm {C}\rightarrow \mathbb {R}^+\) has positive infimum, where \(\mathrm {C}\) is a convex compact subset of \(\mathbb {R}^n\). This implies a constructive separation theorem for convex sets.

Higher-order sensitivity analysis in set-valued optimization under Henig efficiency

The behavior of the perturbation map is analyzed quantitatively by using the concept of higher-order contingent derivative for the set-valued maps under Henig efficiency. By using the higher-order contingent derivatives and applying a separation theorem for convex sets, some results concerning higher-order sensitivity analysis are established.

An Open-loop criterion for the solvability of a closed-loop guidance problem with incomplete information: Linear control systems

The method of open-loop control packages is a tool for stating the solvability of guaranteed closed-loop control problems under incomplete information on the observed states. In this paper, a solution method is specified for the problem of guaranteed closed-loop guidance of a linear control system to a convex target set at a prescribed point in time. It is assumed that the observed signal on the system’s states is linear and the set of its admissible initial states is finite. It is proved that the problem under consideration is equivalent to the problem of open-loop guidance of an extended linear control system to an extended convex target set. Using a separation theorem for convex sets, we derive a solvability criterion, which reduces to solving a finite-dimensional optimization problem. An illustrative example is considered.

SEPARATION OF CONVEX SETS IN EXTENDED NORMED SPACES

We give continuous separation theorems for convex sets in a real linear space equipped with a norm that can assume the value infinity. In such a space, it may be impossible to continuously strongly separate a point $p$ from a closed convex set not containing $p$, that is, closed convex sets need not be weakly closed. As a special case, separation in finite-dimensional extended normed spaces is considered at the outset.

Farkas Lemma for Convex Systems Revisited and Applications to Sublinear-Convex Optimization Problems

In this paper we establish new versions of the Farkas lemma for systems which are convex with respect to a cone and convex with respect to an extended sublinear function under some Slater-type constraint qualification conditions and in the absence of lower semi-continuity and closedness assumptions on the functions and constrained sets. The results can be considered as counterparts of some of the earlier corresponding results in Dinh et al. (SIAM J. Optim. 24: 678–701, 2014). As consequences, we get extensions of the Hahn–Banach theorem for extended sublinear functions (the situation where the celebrated Hahn–Banach theorem failed). The results obtained are then applied to provide duality results and optimality conditions for a class of composite problems involving sublinear-convex mappings. Two special cases are examined at the end of the paper. In the first one our results give rise to some generalized Fenchel duality theorems while in the second one, in normed spaces, our result leads to the separation theorem for convex sets (not necessarily closed) in normed spaces.