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.     

A Solution Concept for Fuzzy Multiobjective Programming Problems Based on Convex Cones

A solution concept for fuzzy multiobjective programming problems based on ordering cones (convex cones) is proposed in this paper. The notions of ordering cones and partial orderings on a vector space are essentially equivalent. Therefore, the optimality notions in a real vector space can be elicited naturally by invoking a concept similar to that of the Pareto-optimal solution in vector optimization problems. We introduce a corresponding multiobjective programming problem and a weighting problem of the original fuzzy multiobjective programming problem using linear functionals so that the optimal solution of its corresponding weighting problem is also the Pareto-optimal solution of the original fuzzy multiobjective programming problem.     

On Solidness of Polar Cones

We investigate the properties of cones whose polars are solid in different polar topologies. By a standard duality argument, we obtain a number of necessary and sufficient conditions for closed convex cones to be solid in various locally convex spaces. From this, we can deduce easily the extensions of previous related results. Furthermore, we construct a class of closed convex cones in some Banach spaces, which are not solid but whose polars satisfy the angle property. This solves the Han conjecture in the negative.     

Cones Admitting Strictly Positive Functionals and Scalarization of Some Vector Optimization Problems

This paper is concerned with cones admitting strictly positive functionals and scalarization methods in multiobjective optimization. Assuming that the ordering cone admits strictly positive functionals or possesses a base in normed spaces or is a supernormal cone in a Banach space, we give scalar and scalar proper representations for vector optimization problems with convex and naturally quasiconvex data.     

An infinite class of convex tangent cones

Since the early 1970's, there have been many papers devoted to tangent cones and their applications to optimization. Much of the debate over which tangent cone is best has centered on the properties of Clarke's tangent cone and whether other cones have these properties. In this paper, it is shown that there are an infinite number of tangent cones with some of the nicest properties of Clarke's cone. These properties are convexity, multiple characterizations, and proximal normal formulas. The nature of these cones indicates that the two extremes of this family of cones, the cone of Clarke and the B-tangent cone or the cone of Michel and Penot, warrant further study. The relationship between these new cones and the differentiability of functions is also considered.     

The area minimizing problem in conformal cones,I

In this paper we study the area minimizing problem in some kinds of conformal cones. This concept is a generalization of the cones in Euclidean spaces and the cylinders in product manifolds. We define a non-closed-minimal (NCM) condition for bounded domains. Under this assumption and other necessary conditions we establish the existence of bounded minimal graphs in mean convex conformal cones. Moreover those minimal graphs are the solutions to corresponding area minimizing problems. We can solve the area minimizing problem in non-mean convex translating conformal cones if these cones are contained in a larger mean convex conformal cones with the NCM assumption. We give examples to illustrate that this assumption can not be removed for our main results.     

Higher-Order Efficiency Conditions Via Higher-Order Tangent Cones

This article presents higher-order necessary and sufficient efficiency conditions for multiobjective optimization problems involving cone-constraints and a set constraint with G\^ateaux differentiable functions via higher-order tangential cones.     

Generalized asymptotic functions in nonconvex multiobjective optimization problems

In this paper, we use generalized asymptotic functions and second-order asymptotic cones to develop a general existence result for the nonemptiness of the proper efficient solution set and a sufficient condition for the domination property in nonconvex multiobjective optimization problems. A new necessary condition for a point to be efficient or weakly efficient solution is given without any convexity assumption. We also provide a finer outer estimate for the asymptotic cone of the weakly efficient solution set in the quasiconvex case. Finally, we apply our results to the linear fractional multiobjective optimization problem.     

Theory of convex cones in multicriteria decision making

We develop the theory of convex polyhedral cones in the objective-function space of a multicriteria decision problem. The convex cones are obtained from the decision-maker's pairwise judgments of decision alternatives and are applicable to any quasiconcave utility function. Therefore, the cones can be used in any progressively articulated solution procedure that employs pairwise comparisons. The cones represent convex sets of solutions that are inferior to known solutions to a multicriteria problem. Therefore, these convex sets can be eliminated from consideration while solving the problem. We develop the underlying theory and a framework for representing knowledge about the decision-maker's preference structure using convex cones. This framework can be adopted in the interactive solution of any multicriteria problem after taking into account the characteristics of the problem and the solution procedure. Our computational experience with different multicriteria problems shows that this approach is both viable and efficient in solving practical problems of moderate size.     

Characterization of Solid Cones

The author gives a dual characterization of solid cones in locally convex spaces. From this the author obtains some criteria for judging convex cones to be solid in various kinds of locally convex spaces. Using a general expression of the interior of a solid cone, the author obtains a number of necessary and sufficient conditions for convex cones to be solid in the framework of Banach spaces. In particular, the author gives a dual relationship between solid cones and generalized sharp cones. The related known results are improved and extended.     

A New Conical Regularization for Some Optimization and Optimal Control Problems: Convergence Analysis and Finite Element Discretization

In this article, we study an abstract constrained optimization problem that appears commonly in the optimal control of linear partial differential equations. The main emphasis of the present study is on the case when the ordering cone for the optimization problem has an empty interior. To circumvent this major difficulty, we propose a new conical regularization approach in which the main idea is to replace the ordering cone by a family of dilating cones. We devise a general regularization approach and use it to give a detailed convergence analysis for the conical regularization as well as a related regularization approach. We showed that the conical regularization approach leads to a family of optimization problems that admit regular multipliers. The approach remains valid in the setting of general Hilbert spaces and it does not require any sort of compactness or positivity condition on the operators involved. One of the main advantages of the approach is that it is amenable for numerical computations. We consider four different examples, two of them elliptic control problems with state constraints, and present numerical results that completely support our theoretical results and confirm the numerical feasibility of our approach. The motivation for the conical regularization is to overcome the difficulties associated with the lack of Slater's type constraint qualification, which is a common hurdle in numerous branches of applied mathematics including optimal control, inverse problems, vector optimization, set-valued optimization, sensitivity analysis, variational inequalities, among others.     

Solutions of Fuzzy Multiobjective Programming Problems Based on the Concept of Scalarization

Scalarization of fuzzy multiobjective programming problems using the embedding theorem and the concept of convex cone (ordering cone) is proposed in this paper. Since the set of all fuzzy numbers can be embedded into a normed space, this motivation naturally inspires us to invoke the scalarization techniques in vector optimization problems to evaluate the a multiobjective programming problem. Two solution concepts are proposed in this paper by considering different convex cones.     

A cone programming approach to the bilinear matrix inequality problem and its geometry

We discuss an approach for solving the Bilinear Matrix Inequality (BMI) based on its connections with certain problems defined over matrix cones. These problems are, among others, the cone generalization of the linear programming (LP) and the linear complementarity problem (LCP) (referred to as the Cone-LP and the Cone-LCP, respectively). Specifically, we show that solving a given BMI is equivalent to examining the solution set of a suitably constructed Cone-LP or Cone-LCP. This approach facilitates our understanding of the geometry of the BMI and opens up new avenues for the development of the computational procedures for its solution. Research supported in part by the National Science Foundation under Grant CCR-9222734.     

Finding solutions of implicit complementarity problems by isotonicity of the metric projection

Isac and Németh [G. Isac and A. B. Németh, Projection method, isotone projection cones and the complementarity problem, J. Math. Anal. App., 153, 258-275(1990)] proved that solving a coincidence point equation (fixed point problem) in turn solves the corresponding implicit complementarity problem (nonlinear complementarity problem) and they exploited the isotonicity of the metric projection onto isotone projection cones to solve implicit complementarity problems (nonlinear complementarity problems) defined by these cones. In this paper, the notion of *-isotone projection cones is employed and an iterative algorithm is presented in connection with an implicit complementarity problem on *-isotone projection cones. It is proved that if the sequence generated through the defined algorithm is convergent, then its limit is a solution of the coincidence point equation and thus solves the implicit complementarity problem. Sufficient conditions are given for this sequence to be convergent for implicit complementarity problems defined by *-isotone projection cones. The question of finding nonzero solutions of these problems is also studied.     

Implicit complementarity problems on isotone projection cones

Isac and Németh [G. Isac and A. B. Németh, Projection methods, isotone projection cones and the complementarity problem, J. Math. Anal. Appl. 153 (1990), pp. 258–275] proved that solving a coincidence point equation (fixed point problem) in turn solves the corresponding implicit complementarity problem (nonlinear complementarity problem) and they exploited the isotonicity of the metric projection onto isotone projection cones to solve implicit complementarity problems (nonlinear complementarity problems) defined by these cones. In this article an iterative algorithm is studied in connection with an implicit complementarity problem. It is proved that if the sequence generated through the defined algorithm is convergent, then its limit is a solution of the coincidence point equation and thus solves the implicit complementarity problem. Sufficient conditions are given for this sequence to be convergent for implicit complementarity problems defined by isotone projection cones, extending the results of Németh [S.Z. Németh, Iterative methods for nonlinear complementarity problems on isotone projection cones, J. Math. Anal. Appl. 350 (2009), pp. 340–370]. Some existing concepts from the latter paper are extended to solve the problem of finding nonzero solutions of the implicit complementarity problem.     

Cone Characterizations of Approximate Solutions in Real Vector Optimization

Approximate nondominated solutions of real vector optimization problems are characterized using the concept of translated cones. Relationships between these solutions and Pareto nondominated points are examined, and the problem of optimizing over the set of approximate solutions is addressed. This research was supported by the National Science Foundation, Grant DMS-0425768, and by the Automotive Research Center, a US Army TACOM Center of Excellence for Modeling and Simulation of Ground Vehicles at the University of Michigan     

Some notes on the paper “Cone metric spaces and fixed point theorems of contractive mappings”

Huang and Zhang reviewed cone metric spaces in 2007 [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468-1476]. We shall prove that there are no normal cones with normal constant M<1 and for each k>1 there are cones with normal constant M>k. Also, by providing non-normal cones and omitting the assumption of normality in some results of [Huang Long-Guang, Zhang Xian, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468-1476], we obtain generalizations of the results.     

A New Self-Dual Embedding Method for Convex Programming

In this paper we introduce a conic optimization formulation to solve constrained convex programming, and propose a self-dual embedding model for solving the resulting conic optimization problem. The primal and dual cones in this formulation are characterized by the original constraint functions and their corresponding conjugate functions respectively. Hence they are completely symmetric. This allows for a standard primal-dual path following approach for solving the embedded problem. Moreover, there are two immediate logarithmic barrier functions for the primal and dual cones. We show that these two logarithmic barrier functions are conjugate to each other. The explicit form of the conjugate functions are in fact not required to be known in the algorithm. An advantage of the new approach is that there is no need to assume an initial feasible solution to start with. To guarantee the polynomiality of the path-following procedure, we may apply the self-concordant barrier theory of Nesterov and Nemirovski. For this purpose, as one application, we prove that the barrier functions constructed this way are indeed self-concordant when the original constraint functions are convex and quadratic. We pose as an open question to find general conditions under which the constructed barrier functions are self-concordant.     

Multiple criteria decision problems with fuzzy domination structures

The concepts of domination structures and nondominated solutions in multiple criteria decision problems, which were introduced by Yu, enable us to tackle general situations in which there exists information concerning the decision maker's preferences.In many of the multiple criteria decision problems the underlying domination structures are not known precisely but only fuzzily determined. Yu primarily works with the case where the domination structure at each point is a convex cone. As a result, there exists a sharp borderline dividing all solutions into nondominated solutions and the others.This paper fuzzifies the concepts of domination structures and nondominated solutions to allow them to be applied to a larger class of the multiple criteria decision problems mentioned above. Introducing the concepts of fuzzy convex cones and fuzzy polar cones, it is shown how some of the main results obtained by Yu are extended.     

Isotone retraction cones in Hilbert spaces

A mapping is called isotone if it is monotone increasing with respect to the order induced by a pointed closed convex cone. Finding the pointed closed convex generating cones for which the projection mapping onto the cone is isotone is a difficult problem which was analyzed in Isac and Németh (1986, 1990, 1992) [1], [2], [3], [4] and [5]. Such cones are called isotone projection cones. In particular it was shown that any isotone projection cone is latticial (Isac (1990) [2]). This problem is extended by replacing the projection mapping with continuous retractions onto the cone. By introducing the notion of sharp mappings, it is shown that a pointed closed convex generating cone is latticial if and only if there is a continuous retraction onto the cone whose complement is sharp. Several particular cases are considered and examples are given.