Vector-valued Objective Function Research Articles

A new approach to zero duality gap of vector optimization problems using characterizing sets

In this paper we propose results on zero duality gap in vector optimization problems posed in a real locally convex Hausdorff topological vector space with a vector-valued objective function to be minimized under a set and a convex cone constraint. These results are then applied to linear programming.

Prioritized Inverse Kinematics: Generalization

This letter proposes a generalization of the prioritized inverse kinematics (PIK) problem as the multi-objective optimization with the lexicographical ordering. We specify three properties for a vector-valued objective function to be proper for the PIK problem, so that the set of all PIK solutions can be generated from the set of all proper objective functions. The dependence property requires that higher priority tasks do not constrain lower priority tasks unnecessarily, the uniqueness property demands that there exists one and only one PIK solution given a proper objective function, and the representation property asks that if there are two distinct references that are realizable by a mechanism, then the PIK solutions for those references should differ. We also include the preconditioning of the velocity mapping functions in our generalization that can be used to handle the numerical imbalance in the orthogonalization that raises a difficulty in choosing damping functions and degenerates the performance of lower priority tasks. We justify our generalization by showing that it discards trivial solutions such as a constant function that is not intended and contains several PIK solutions including two successful PIK solutions, so-called Nakamura's and (weighted) Chiaverini's solutions, with and without damping. We compare those PIK solutions by a simulation with a seven-degree-of-freedom manipulator, KUKA-LWR.

Ekeland variational principles for set-valued functions with set perturbations

By using generalized nonconvex separation functionals and a pre-order principle in Qiu [J Math Anal Appl. 2014;419:904–937], we establish a general set-valued Ekeland variational principle (briefly, denoted by EVP), where the objective function is a set-valued map taking values in a real vector space quasi-ordered by a convex cone K and the perturbation consists of a cone-convex subset H of K multiplied by the distance function. Here, the assumption on lower semi-continuity of the objective function is replaced by a weaker one: sequentially lower monotony. And the assumption on lower boundedness of the objective function is taken to be the weakest of several different kinds. From the general set-valued EVP, we deduce a number of particular versions of set-valued EVP, which extend and improve the related results in the literature. In particular, we give several EVPs for approximately efficient solutions in set-valued optimization, which not only extend the related results from vector-valued objective functions into set-valued objective functions, but also improve the related results by removing a usual assumption for K-boundedness (by scalarization) of the objective function's range.

Cost approach methodology in assessing economic stability of enterprise of hotel and restaurant sector

The article is devoted to improving the employment of cost approach methodology in assessing economic stability of enterprise of hotel and restaurant sector. The essence and the procedure of cost approach which allows calculating qualitative and quantitative value of every cost element and define its relative share in the total resource intensity have been investigated in the paper. It is determined that the most efficient strategic instrument for solving the issues of ensuring economic stability of the enterprise in hotel and restaurant sector is the vector-valued objective function. Its primary goal is to determine the object optimal for each specific criterion. Proceeding from the objective of ensuring sustainable economic growth, vector-valued objective function will include the assessment criteria – performance indicators and resource intensity indicators of operational activity of enterprise that maximize or minimize. The identified criteria assess various states of the analyzed object, with regard to which the decisions are approved. It has been substantiated that the use of economic stability assessment in the taxonomy on the basis of cost approach allows aggregating diversified indicators of the enterprise into a single indicator and identify the impact of various factors upon efficiency and, consequently, stability of the enterprise activity. It is determined that within the scope of the method of calculation by unit costs indicators between resultrelated and factor-related parameters the connection is employed in the form of proportional linear function. Such approximation cab be maintained only within narrow range of variation of resource intensity indicator. Therefore, for each range it is expedient to allocate its specific ‘per unit’ indicator. Assessment of dynamic economic sustainability of the enterprise is forecasted within a particular time period by means of determining residual variance or root-mean-square deviation indicator (sample variance). An algorithm is established with regard to the agenda for action that encompasses analysis, assessment, forecasting, optimization and serves as an instrument of managing the stability of the enterprise based upon cost monitoring and objective function. Realization of the object function with set restrictions contributes to forming the mechanism of sustainable development of the enterprise in the conditions of uncertainty and dynamic nature of external environment.

Optimality conditions for nonsmooth multiobjective bilevel optimization problems

This article is devoted to the study of a nonsmooth multiobjective bilevel optimization problem, which involves the vector-valued objective functions in both levels of the considered program. We first formulate a relaxation multiobjective formulation for the multiobjective bilevel problem and examine the relationships of solutions between them. We then establish Fritz John (FJ) and Karush–Kuhn–Tucker (KKT) necessary conditions for the nonsmooth multiobjective bilevel optimization problem via its relaxation. This is done by studying a related multiobjective optimization problem with operator constraints.

Operational Planning in Power Systems Based on Multi-Objective Optimization

Acharacteristic feature of an electric power system (EPS) is a great number of possible states. Each state is described by a set of independent parameters representing reliability, stability, efficiency, etc. It is proposed to use a vector-valued objective function to assess a state for acceptability. The set of best states can be found using multi-objective optimization, which is considered as a method to make effective decisions.

First and second order necessary optimality conditions for a discrete-time optimal control problem with a vector-valued objective function

The paper deals with a nonlinear discrete-time optimal control problem with a vector-valued objective function of terminal type. For the admissible controls of the problem under consideration which satisfy the additional regularity condition of the Lyusternik type we prove the first and second necessary optimality conditions extending such the classical optimality conditions as the Euler condition and the negativity of second variation of the objective function.

A hybrid evolutionary algorithm for distribution feeder reconfiguration

Distribution feeder reconfiguration (DFR) is formulated as a multi-objective optimization problem which minimizes real power losses, deviation of the node voltages and the number of switching operations and also balances the loads on the feeders. In the proposed method, the distance (λ 2 norm) between the vector-valued objective function and the worst-case vector-valued objective function in the feasible set is maximized. In the algorithm, the status of tie and sectionalizing switches are considered as the control variables. The proposed DFR problem is a non-differentiable optimization problem. Therefore, a new hybrid evolutionary algorithm based on combination of fuzzy adaptive particle swarm optimization (FAPSO) and ant colony optimization (ACO), called HFAPSO, is proposed to solve it. The performance of HFAPSO is evaluated and compared with other methods such as genetic algorithm (GA), ACO, the original PSO, Hybrid PSO and ACO (HPSO) considering different distribution test systems.

Generalized (F,ρ)-Convexity and Duality in Nonsmooth Problems of Multiobjective Optimization

A mixed-type dual for a nonsmooth multiobjective optimization problem with inequality and equality constraints is formulated. We obtain weak and strong duality theorems for a mixed-type dual without requiring the regularity assumptions and the nonnegativeness of the Lagrange multipliers associated to the equality constraints. We apply also a nonsmooth constraint qualification for multiobjective programming to establish strong duality results. In this case, our constraint qualification assures the existence of positive Lagrange multipliers associated with the vector-valued objective function.

Evaluating ecological tradeoffs in fisheries management: a study case for the yellowfin tuna fishery in the Eastern Pacific Ocean

Multiobjective decision analysis (MDA) is a useful assessment method when fishery managers need a systematic investigation of the tradeoffs involved in the selection of alternative policy options. An important class of techniques within MDA is vector optimization, consisting of mathematical programming models with vector valued objective functions. From the management perspective, vector optimization models are particularly suited for situations when the decision rule requires each objective to be kept as high (or low) as possible. Solving vector optimization problems usually entails finding a set of Pareto-optimal solutions. These solutions are relevant to the decision-making process specially if decision-makers have monotonic preferences. In this paper, a vector optimization model of the Eastern Pacific yellowfin tuna fishery is used to generate Pareto-optimal solutions and to evaluate the tradeoffs (shadow prices) involved in the selection of alternative policy options. Three conflicting biological objectives are considered: (a) minimizing dolphin incidental mortality, (b) minimizing by-catch of all non-dolphin species and (c) maximizing total yellowfin tuna catch. Results are presented and discussed by means of non-linear tradeoff curves.

On the convergence of multiobjective evolutionary algorithms

We consider the usage of evolutionary algorithms for multiobjective programming (MOP), i.e. for decision problems with alternatives taken from a real-valued vector space and evaluated according to a vector-valued objective function. Selection mechanisms, possibilities of temporary fitness deterioration, and problems of unreachable alternatives for such multiobjective evolutionary algorithms (MOEAs) are studied. Theoretical properties of MOEAs such as stochastic convergence with probability 1 are analyzed.

A generalized upper dini-directional derivation in vector optimization

In this paper, a set-valued generalized upper Dini-directional derivative is introduced for a locally lipschitz vector-valued function. Some properties, such as sum formula and chain rule, of this upper Dini-directional derivative are derived. This upper Dini-directional derivative is applied to characterize a cone-convex function and a vector subdifferential and to derive optimality conditions for a multi-objective optimization problem with a locally Lipschitz vector-valued objective function over a convex set.

On variational principles, level sets, well-posedness, and ∈-solutions in vector optimization

Using the concept for ∈-efficient solutions introduced in Refs. 1 and 2, we extend the Ekeland variational principle and another variational principle given in Ref. 3 to vector-valued objective functions. This enables us to establish a kind of well-posedness for the resulting perturbed vector optimization problems. Based on this definition of ∈-efficiency, we also formulate a concept for level sets and prove some results about the Kuratowski limits of level sets.

Optimal design using min - max criteria for two-dimensional recursive digital filters

This paper concerns design techniques for two-dimensional recursive digital filters in the frequency domain. The filters are designed so as to approximate simultaneously prescribed specifications of magnitude and group delay by treating the coefficients of the transfer function as decision variables. These approximation problems are formulated as minimizing the errors between the filter's response and the desired one with respect to the magnitude and group delay characteristics. The following two types of optimization techniques are proposed: (1) the Min-Max Type Optimization Satisfaction Method and (2) the Min-Max Type Multi-Objective Optimization Method. The former is a method that optimizes (approximates) the magnitude characteristic in a min-max sense under the condition requiring that the error of group delay should be kept below the tolerance level. The latter is a method that optimizes the vector-valued objective function of the errors of magnitude and group delay. The optimal solution for this multiobjective problem is obtained by combining the satisfaction approach and the maximum component minimization technique for vector optimization. Since these problems include maximum-valued functions in the performance indices, as a nondifferentiable optimization technique, we apply Mifflin's algorithm using generalized gradients. Several numerical examples are presented.

Duality and reciprocity for vector programming

Starting from the well-known result that the Lagrange multipliers measure the sensitivity of the optimum-value function with respect to the constraint parameters, we develop a generalized duality theory for vector programming problems. Then we consider the connection, stated by the complementarity property, of the optimum-value function with the saturation of the constraints and introduce “reciprocal problems” with many constraints and a vector-valued objective function. For such problems we derive some monotonicity properties, significantly related to the active constraints and apply these results to the sensitivity analysis of economic problems.

Strong Vector Minimization and Duality

AbstractA constrained minimum of a vector valued function is defined, in terms of an ordering cone. The definition chosen leads to a vector analog of the Kuhn‐Tucker theorem, and to a duality theorem where the dual problem has a vector valued objective function, and weak duality is defined by appropriate cone orderings. Also proved are a vector valued converse duality theorem, and a vector quasiduality theorem which does not require convexity. The results are related to perturbations of the objective function in a minimization problem.

Multiobjective dynamic programming

In this paper the method of dynamic programming is carried to recurrence computations of sets of optimal values for discrete dynamic systems and vector-valued objective functions.

Vektorwertige dynamische optimierang bei vollständiger beziehungsweise unvollständiger information fiber zeitabhänglge einfluβparameter

Multistep decision processes with a vector-valued objective function are considered whereby the information about the step-dependent parameters is either complete or incomplete. In case of complete information the well-known necessary and sufficient functional equations can be extended with respect to the parameter dependence, but in case of incomplete information only the sufficiency of certain functional equations can be proved.

Necessary Optimality Conditions with Application to a Variational Problem

For an abstract optimization problem with operator constraints and vector-valued objective function, necessary optimality conditions of Kuhn–Tucker type are developed, and are then used to obtain a generalized Euler–Lagrange condition for the problem of Bolza with phase restrictions in the nonconvex and nondifferentiable case.

The Optimization of Systems with Vector-Valued Objective Function

The optimization problem of systems with non-scalar objective function is considered. The notions of p-convex sets and p-convex functions, 1 ≤ p +∞, which coincide for p=1 with the known notion of convexity are introduced. When supposed that components of objective function are p-convex and the set of admissible vectors is convex the necessary and sufficient condition is proved. For the convex problem of vector optimization the special equation is constructed. The set of noninferior roots of this equation coincides with the set of noninferior values of objective function.