Vector Maximization Problem Research Articles

Solution of the general multi-objective De-Novo programming problem using compensatory operator under fuzzy environment

This paper presents an alternative approach for the solution of the general multi-objective De-Novo Programming Problem under fuzzy environment in one step using Luhandjula’s compensatory µθ-operator. Luhandjula used compensatory µθ-operator to solve vector maximization problem, whereas in this paper the use of compensatory µθ-operator has been extended to solve the general multi-objective De-Novo Programming Problem. Also the solution obtained by the proposed approach elicits an efficient solution of the problem considered under the assumption of the uniqueness of the solution. The method has been illustrated by a numerical example.

On Artificial-Noise-Aided Transmit Design for Multiuser MISO Systems With Integrated Services

This paper considers artificial noise (AN)-aided transmit designs for multi-user MISO systems in the eyes of service integration. Specifically, we combine two sorts of services, and serve them simultaneously: one multicast message intended for all receivers and one confidential message intended for only one receiver. The confidential message is kept perfectly secure from all the unauthorized receivers. Our goal is to jointly design the optimal input covariances for the multicast message, confidential message and AN, such that the achievable secrecy rate region is maximized subject to the sum power constraint. This secrecy rate region maximization (SRRM) problem is a nonconvex vector maximization problem. To handle it, we reformulate the SRRM problem into a provably equivalent scalar optimization problem and propose a searching method to find all of its Pareto optimal points. The equivalent scalar optimization problem is identified as a secrecy rate maximization (SRM) problem with the quality of multicast service (QoMS) constraints. Further, we show that this equivalent QoMS-constrained SRM problem, albeit nonconvex, can be efficiently handled based on a two-stage optimization approach, including solving a sequence of semidefinite programs. Moreover, we also extend the SRRM problem to an imperfect channel state information (CSI) case where a worst-case robust formulation is considered. In particular, while transmit beamforming is generally a suboptimal technique to the SRRM problem, we prove that it is optimal for the confidential message transmission whether in the perfect CSI scenario or in the imperfect CSI scenario. Finally, numerical results demonstrate that the AN-aided transmit designs are effective in expanding the achievable secrecy rate regions.

Energy Efficiency Region for Gaussian MISO Channels With Integrated Services

This letter studies energy efficient optimization problem for a two-receiver multiple-input single-output channel with two integrated service messages: one multicast message intended for both receivers and one confidential message intended only for one receiver and required to be perfectly secure from the other receiver. We aim to characterize the energy efficiency (EE) tradeoff between the two services, and more specifically, to find the Pareto boundary of the EE region. Such an EE region maximization problem is a nonconvex vector maximization problem, and we propose a method of scalarization to reformulate it into a scalar optimization problem. Despite the nonconvexity of the scalar problem, we show that it can be iteratively solved by combining fractional programming and difference-of-concave programming methods. Further, we prove that our obtained EE performance could always be achieved by simply using the single-stream beamforming for confidential message transmission. Finally, numerical results demonstrate the efficacy of our proposed methods in characterizing the EE tradeoff.

Conjugate Duality and Optimization over Weakly Efficient Set

In this article, we present a conjugate duality for nonconvex optimization problems. This duality scheme is symmetric and has zero gap. As applied to a vector-maximization problem, it transforms the latter into an optimization problem over a weakly efficient set which can be solved by monotonic optimization methods.

Secrecy Capacity Region Maximization in Gaussian MISO Channels with Integrated Services

This letter considers a two-receiver multiple-input single-output Gaussian broadcast channel model with integrated services. Specifically, two sorts of service messages are combined and served simultaneously: one multicast message intended for both receivers and one confidential message intended for only one receiver. The confidential message is kept perfectly secure from the unauthorized receiver. Our goal is to jointly design the input covariances for the multicast message and confidential message, such that the secrecy capacity region is maximized. This secrecy capacity region maximization (SCRM) problem is a nonconvex vector maximization problem. To deal with this issue, we reformulate the SCRM problem into a provably equivalent scalar optimization problem and propose a searching method to find its overall Pareto optimal points. Further, for implementation efficiency, transmit beamforming is proved to be Pareto optimal. However, since the two service messages are coupled in our optimization problem, it is difficult to deduce closed-form expressions of the Pareto optimal beamformers. A suboptimal transmit design is accordingly proposed to analytically obtain beamformers for both service messages. Numerical results illustrate that the performance gap between the Pareto optimal design and our proposal is negligible.

Problems with resource allocation constraints and optimization over the efficient set

The paper studies a nonlinear optimization problem under resource allocation constraints. Using quasi-gradient duality it is shown that the feasible set of the problem is a singleton (in the case of a single resource) or the set of Pareto efficient solutions of an associated vector maximization problem (in the case of $$k>1$$ resources). As a result, a nonlinear optimization problem under resource allocation constraints reduces to an optimization over the efficient set. The latter problem can further be converted into a quasiconvex maximization over a compact convex subset of $$\mathbb{R }^k_+.$$ Alternatively, it can be approached as a bilevel program and converted into a monotonic optimization problem in $$\mathbb{R }^k_+.$$ In either approach the converted problem falls into a common class of global optimization problems for which several practical solution methods exist when the number $$k$$ of resources is relatively small, as it often occurs.

A Mountain Pass-type Theorem for Vector-valued Functions

The mountain pass theorem for scalar functionals is a fundamental result of the minimax methods in variational analysis. In this work we extend this theorem to the class of $\mathcal{C}^{1}$ functions $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ , where the image space is ordered by the nonnegative orthant $\mathbb{R}_{+}^{m}$ . Under suitable geometrical assumptions, we prove the existence of a critical point of f and we localize this point as a solution of a minimax problem. We remark that the considered minimax problem consists of an inner vector maximization problem and of an outer set-valued minimization problem. To deal with the outer set-valued problem we use an ordering relation among subsets of $\mathbb{R}^{m}$ introduced by Kuroiwa. In order to prove our result, we develop an Ekeland-type principle for set-valued maps and we extensively use the notion of vector pseudogradient.

Conjugate duality for vector-maximization problems

In this article we present a conjugate duality for a problem of maximizing a polyhedral concave nondecreasing homogeneous function over a convex feasible set in the nonnegative n-dimensional orthant. Using this duality we obtain a zero-gap duality for a vector-maximization problem.

Linear variational principle for convex vector maximization

For infinitely dimensional convex vector maximization problems, it is proved that there exists, under certain conditions, an arbitrarily small additive perturbation of the performance criterion by a linear continuous operator such that the perturbed problem has efficient solutions.

A New Duality Approach to Solving Concave Vector Maximization Problems

We introduce a special class of monotonic functions with the help of support functions and polar sets, and use it to construct a scalarized problem and its dual for a vector optimization problem. The dual construction allows us to develop a new method for generating weak efficient solutions of a concave vector maximization problem and establish its convergence. Some numerical examples are given to illustrate the applicability of the method.

Unbounded Components in the Solution Sets of Strictly Quasiconcave Vector Maximization Problems

Let (P) denote the vector maximization problem $$\max\{f(x)=\big(f_1(x),\ldots,f_m(x)\big){:}\,x\in D\},$$ where the objective functions f i are strictly quasiconcave and continuous on the feasible domain D, which is a closed and convex subset of R n . We prove that if the efficient solution set E(P) of (P) is closed, disconnected, and it has finitely many (connected) components, then all the components are unbounded. A similar fact is also valid for the weakly efficient solution set E w (P) of (P). Especially, if f i (i=1,...,m) are linear fractional functions and D is a polyhedral convex set, then each component of E w (P) must be unbounded whenever E w (P) is disconnected. From the results and a result of Choo and Atkins [J. Optim. Theory Appl. 36, 203---220 (1982.)] it follows that the number of components in the efficient solution set of a bicriteria linear fractional vector optimization problem cannot exceed the number of unbounded pseudo-faces of D.

Optimality Conditions for Vector Optimization Problems of a Difference of Convex Mappings

In this paper, we study optimality conditions for vector optimization problems of a difference of convex mappings $$ (VP)\left\{\matrix{{\mathbb R}_{+}^{p} &-\hbox{Minimize} f(x)-g(x),\cr &\hbox{subject to the constraints} \cr & x\in C,l(x)\in -Q,Ax=b\cr &\hbox{and } h(x)-k(x)\in -{\mathbb R}_{+}^{m},}\right. $$ where \(f:=(f_{1},\ldots,f_{p}),g:=(g_{1},\ldots,g_{p}) h:=(h_{1},\ldots,h_{m}), k:=(k_{1},\ldots,k_{m}), Q\) is a closed convex cone in a Banach space Z, l is a mapping Q-convex from a Banach space X into Z, A is a continuous linear operator from X into a Banach space \(W, {\mathbb R}_{+}^{p}\) and \({\mathbb R}_{+}^{m}\) are respectively the nonnegative orthants of \({\mathbb R}^{p}\) and \({\mathbb R}^{m}\), C is a nonempty closed convex subset of X, b∈W, and the functions f i ,g i ,h j and k j are convex for i=1,...,p and j=1,ldots,m. Necessary optimality conditions for (VP) are established in terms of Lagrange-Fritz-John multipliers. When the set of constraints for (VP) is convex and under the generalized Slater constraint qualification introduced in Jeyakumar and Wolkowicz [11] , we derive necessary optimality conditions in terms of Lagrange-Karush-Kuhn-Tucker multipliers which are also sufficient whenever the functions g i ,i=1,...,p are polyhedrals. Our approach consists in using a special scalarization function. A necessary optimality condition for convex vector maximization problem is derived. Also an application to vector fractional mathematical programming is given. Our contribution extends the results obtained in scalar optimization by Hiriart-Urruty [9] and improve substantially the few results known in vector case (see for instance: [11], [12] and [14]).

Contractibility of the Solution Sets in Strictly Quasiconcave Vector Maximization on Noncompact Domains

We study the contractibility of the efficient solution set of strictly quasiconcave vector maximization problems on (possibly) noncompact feasible domains. It is proved that the efficient solution set is contractible if at least one of the objective functions is strongly quasiconcave and any intersection of level sets of the objective functions is a compact (possibly empty) set. This theorem generalizes the main result of Benoist (Ref.1), which was established for problems on compact feasible domains.

Linear fractional vector optimization problems with many components in the solution sets

Linear fractional vector optimization (LFVO) problems form a special class of nonconvex multiobjective optimization problems which has a significant role both in the management science and in the theory of vector optimization. Up to now, only LFVO problems with at most two connected components in the solution sets have been discussed in the literature. We propose some examples of LFVO problems with three or more connected components in the solution sets. It is proved that for any integer $m$ there exist LFVO problems with $m$ objective criteria whose solution sets have exactly $m$ connected components. Besides, we have solved the conjecture saying that $\chi(E(\mbox{P}))\leq \min\{m,\mbox{dim}0^+D+1\},$ where $\chi(E(\mbox{P}))$ is the number of connected components in the efficient solution set of a LFVO problem $(\mbox{P})$, $m$ is the number of the objective criteria of $(\mbox{P})$, and $\mbox{dim}0^+D$ is the dimension of the recession cone $0^+D$ of the feasible domain $D$ of $(\mbox{P})$. These new facts are useful for analyzing the practical problems which can be modeled as quasiconcave vector maximization problems in general, and as LFVO problems on unbounded feasible domains in particular.

On computational solution of vector maximum problem

In this paper, we associate a vector maximum optimization (VOP) problem to the related real valued programming problem depending on the following approaches, the minimum operator and the second is product operator. For the above two operators we use a suitable transformation functions to reformulate the (VOP) to the fuzzy problem (FP). A characterization of this transformation functions of the both approaches will be given. We present the relation between the efficient solution and solution of VOP and FP by the minimum and product operator.

Stopped Markov decision processes with multiple constraints

In this paper, a optimization problem for stopped Markov decision processes with vector-valued terminal reward and multiple running cost constraints is considered. Applying the idea of occupation measures and using the scalarization technique for vector maximization problems we obtain the equivalent Mathematical Programming problem and show the existence of a Pareto optimal pair of stationary policy and stopping time requiring randomization in at most k states, where k is the number of constraints. Moreover Lagrange multiplier approaches are considered. The saddle-point statements are given, whose results are applied to obtain a related parametric Mathematical Programming, by which the problem is solved. Numerical examples are given.

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 $$g = (g_1 ,...,g_n )$$ and a convex compact set D in ℝ 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.

Connectedness of the Efficient Set for Strictly Quasiconcave Sets

Given a closed subset X in , we show the connectedness of its efficient points or nondominated points when X is sequentially strictly quasiconcave. In the particular case of a maximization problem with n continuous and strictly quasiconcave objective functions on a compact convex feasible region of , we deduce the connectedness of the efficient frontier of the problem. This work solves the open problem of the efficient frontier for strictly quasiconcave vector maximization problems.

Second order optimality conditions in multiobjective programming

In this paper we will establish some necessary and or sufficient optimality conditions for a vector maximization problem. These conditions are firstly stated without the use of any constraint qualifications. then the given approach allow us to introduce two different kinds of regularity conditions. The image space approach allow us also to state some sufficient optimality conditions in a general form

On the redundancy of convex non-decreasing transformations in vector maximum problems

On the redundancy of convex non-decreasing transformations in vector maximum problems