Search

Their work is an extension of algorithms devised for equilibrium problems, proposed by Moudafi, and by Iusem and Sosa, to the more general context of quasi-equilibrium problems. We are very proud to present the special issue of Optimization related to the XIII Brazilian Workshop on Continuous Optimization, held in September 2019, celebrating Alfredo Iusem's 70th anniversary. Pedro Jorge S. Santos, Paulo Sérgio M. Santos, and Susana Scheimberg contributed their paper I A Newton-type method for Quasi-Equilibrium Problems and applications i , where they extend a Newton-type method for particular Equilibrium Problems to the setting of Quasi-Equilibrium Problems. [Extracted from the article]

Projections onto sets are used in a wide variety of methods in optimization theory but not every method that uses projections really belongs to the class of projection methods as we mean it here. Here,projection methodsare iterative algorithms that use projections onto sets while relying on the general principle that when a family of (usually closed and convex) sets is present, then projections (or approximate projections) onto the given individual sets are easier to perform than projections onto other sets (intersections, image sets under some transformation, etc.) that are derived from the given family of individual sets. Projection methods employ projections (or approximate projections) onto convex sets in various ways. They may use different kinds of projections and, sometimes, even use different projections within the same algorithm. They serve to solve a variety of problems which are either of the feasibility or the optimization types. They have different algorithmic structures, of which some are particularly suitable for parallel computing, and they demonstrate nice convergence properties and/or good initial behavioural patterns. This class of algorithms has witnessed great progress in recent years and its member algorithms have been applied with success to many scientific, technological and mathematical problems. This annotated bibliography includes books and review papers on, or related to, projection methods that we know about, use and like. If you know of books or review papers that should be added to this list please contact us. [ABSTRACT FROM AUTHOR]

Metric regularity is widely concerned since its important applications in optimization and control theory. For promoting the application of metric regularity, it is valuable to study the estimation of the neighborhood which makes the regularity hold. However, it seems that no result has been established about this issue. This paper investigates the estimation of the neighborhood of metric regularity for quadratic functions. The main result gives the expression of the neighborhood of metric regularity for a kind of convex quadratic functions. [ABSTRACT FROM AUTHOR]

This note concerns the paper [Janiak A, Kovalyov MY, Lichtenstein M. On a single machine-scheduling problem with separated position and resource effects. Optimization; 2013. doi:10.1080/02331934.2013.804077], which presents an analysis, a counterexample and a pseudocode related with our proof of optimality for a resource allocation algorithm given in [Rudek A, Rudek R. A note on optimization in deteriorating systems using scheduling problems with the aging effect and resource allocation models. Comput. Math. Appl. 2011;62:1870–1878]. We show that the discussed analysis is based only on one part of our proof omitting its integral second part, which is the source of misunderstanding. The considered counterexample is applied for an algorithm, which was not the method presented in our paper, whereas our algorithm provides the correct result for the mentioned counterexample. The provided pseudocode of the resource allocation algorithm, which is presented as the correct method, is a pseudocode of the algorithm described in our paper. Therefore, we show that the results presented in our paper are correct. [ABSTRACT FROM AUTHOR]

In this paper, we study a vector optimization problem (VOP) with both inequality and equality constraints. We suppose that the functions involved are Fréchet differentiable and their Fréchet derivatives are continuous or stable at the point of study. By virtue of a second-order constraint qualification of Abadie type, we provide second-order Karush-Kuhn-Tucker type necessary optimality conditions for the VOP. Moreover, we also obtain second-order sufficient optimality conditions for a kind of strict local efficiency. Both the necessary conditions and the sufficient conditions are shown in equivalent pairs of primal and dual formulations by using theorems of the alternative for the VOP. [ABSTRACT FROM AUTHOR]

The solution existence of finite horizon optimal economic growth problems is studied by invoking Filippov's Existence Theorem for optimal control problems from the monograph of Cesari L. [Optimization theory and applications. New York: Springer-Verlag; 1983]. Our results are obtained not only for general problems but also for typical ones, where the production function is given by either the AK function or the Cobb–Douglas one, while the utility function can be in a linear or power form. Some open questions and conjectures about the regularity of global solutions of finite horizon optimal economic growth problems are formulated in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we propose new numerical algorithms in the setting of unconstrained optimization problems and we prove the discrete rate of convergence of order O 1 / n 2 in the iterates of the convex objective function. Our optimization algorithms are obtained via discretizations from dynamical systems with Hessian-driven damping. Finally, some numerical experiments are presented in order to validate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper contains a generalized Gordan-type alternative theorem for set-valued maps which characterizes set relations without any convexity assumptions using certain evaluation functions. As a direct consequence and as a good example, we discuss robustness (or stability) of linear programming problems for modelling error. Moreover, this theorem can be utilized for that of general vector optimization problems in special cases due to reformation of the evaluation functions. [ABSTRACT FROM AUTHOR]

In this article, a class of lower-order smoothed objective penalty functions is introduced to find locally optimal points for constrained optimization problems. The exactness of the new penalty functions is studied. Based on the current locally optimal points, a new class of penalty functions based on filling properties is proposed. This new penalty function can be used to find a better locally optimal point. The exactness and filling properties of this penalty function are proved in this paper. To do this, two algorithms are presented to find the locally and globally optimal points. Additionally, their convergence is proved under some mild conditions. Finally, numerical results are included to illustrate the applicability of the local and global optimization algorithms. [ABSTRACT FROM AUTHOR]

In this paper, a class of linear multiplicative problems (LMP) are considered, which cover many applications and are known to be NP-hard. For finding the globally optimal solution to problem (LMP) with a pre-specified ε-tolerance, problem (LMP) is first transformed into an equivalent problem (EP) via introducing the variable transformation. And, a novel linear relaxation technique is presented by exploiting the special structure of problem (EP), for deriving the linear relaxation programming which can be used to acquire the upper bound of the optimal value to problem (EP). A branch and bound algorithm is then located for globally solving problem (LMP). The convergence of the algorithm is established and its computational complexity is estimated. Finally, numerical results are reported to illustrate the feasibility and efficiency of the proposed algorithm. [ABSTRACT FROM AUTHOR]

In this paper, proper minimal elements of a given nonconvex set in a real ordered Banach space are defined utilizing the limiting (Mordukhovich) normal cone. The newly defined points are called limiting proper minimal (LPM) points. It is proved that each LPM is a proper minimal in the sense of Borwein under some assumptions. The converse holds in Asplund spaces. The relation of LPM points with Benson, Henig, super and proximal proper minimal points are established. Under appropriate assumptions, it is proved that the set of robust elements is a subset of the set of LPM points, and the set of LPM points is dense in that of minimal points. Another part of the paper is devoted to scalarization-based and distance function-based characterizations of the LPM points. The paper is closed by some results about LPM solutions of a set-valued optimization problem via variational analysis tools. Clarifying examples are given in addition to the theoretical results. [ABSTRACT FROM AUTHOR]

Generalized equations are problems emerging in contexts of modern variational analysis as an adequate formalism to treat issues such as constraint systems, optimality and equilibrium conditions, variational inequalities, differential inclusions. This paper contains a study on solvability and error bounds for generalized equations of the form , where F is a given set-valued mapping and C is a closed, convex cone. A property called metric C-increase, matching the metric behaviour of F with the partial order associated with C, is singled out, which ensures solution existence and error bound estimates in terms of problem data. Applications to the exact penalization of optimization problems with constraint systems, defined by the above class of generalized equations, and to the existence of ideal efficient solutions in vector optimization are proposed. [ABSTRACT FROM AUTHOR]

This paper is a continuation of our effort in using mathematical optimization involving DC programming in clustering and multifacility location. We study a penalty method based on distance functions and apply it particularly to a number of problems in clustering and multifacility location in which the centers to be found must lie in some given set constraints. We also provide numerical examples to test our method. [ABSTRACT FROM AUTHOR]

In this paper, some nonsmooth analogues of the Burachik-Rizvi regularity conditions are introduced for a nonsmooth multiobjective optimization problem with equality and inequality constraints and with an arbitrary set constraint. Primal first-order necessary conditions for local efficiency and local Geoffrion-proper efficiency are given for this problem. Dual Kuhn-Tucker necessary conditions for local efficiency and local Geoffrion-proper efficiency are obtained for the multiobjective problem with equality and inequality constraints. [ABSTRACT FROM AUTHOR]

The non-negative regularization problem has been widely studied for finding non-negative sparse solutions of linear inverse problems and gained successful applications in various application areas. In the present paper, we propose an iterative positive thresholding algorithm (IPTA) to solve the non-negative regularization problem and investigate its convergence properties in finite- or infinite-dimensional Hilbert spaces. The significant advantage of the IPTA is that it is very simple and of low computation cost, and thus, it is practically attractive, especially for large-scale problems. The global convergence of the IPTA is achieved under some mild assumptions on algorithmic parameters. Furthermore, we introduce a notion of positive orthogonal sparsity pattern, and use it to establish the linear convergence rate of the IPTA to a global minimum. Finally, the numerical study on compressive sensing shows that the proposed IPTA is efficient in approaching the non-negative sparse solutions of linear inverse problems and outperforms several existing algorithms in sparse optimization. [ABSTRACT FROM AUTHOR]

In this paper, we construct appropriate aggregate mappings and a new aggregate constraint homotopy (ACH) equation by converting equality constraints to inequality constraints and introducing two variable parameters. Then, we propose an ACH method for nonlinear programming problems with inequality and equality constraints. Under suitable conditions, we obtain the global convergence of this ACH method, which makes us prove the existence of a bounded smooth path that connects a given point to a Karush-Kuhn-Tucker point of nonlinear programming problems. The numerical tracking of this path can lead to an implementable globally convergent algorithm. A numerical procedure is given to implement the proposed ACH method, and the computational results are reported. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonsmooth vector optimization problem involving locally Lipschitz generalized approximate convex functions and find some relations between approximate convexity and generalized approximate convexity. We establish relationships between vector variational inequalities and nonsmooth vector optimization problem using the generalized approximate convexity as a tool. [ABSTRACT FROM AUTHOR]

This paper focuses on how to compare two fuzzy sets and, from the viewpoint of set optimization, proposes eight types of fuzzy-set relations based on a convex cone as new comparison criteria of fuzzy sets. Then, difference evaluation functions for fuzzy sets are introduced. Under suitable assumptions of certain compactness and stability of fuzzy sets, we show that these functions correspond well to the fuzzy-set relations. In addition, through transforming these functions stepwise, we deal with numerical calculation methods of them in particular cases. Consequently, we can judge whether each fuzzy-set relation holds or not for given two fuzzy sets with the aid of computers. [ABSTRACT FROM AUTHOR]

Numerous empirical studies show that portfolio returns are generally asymmetric. In this paper, skewness is considered to measure the asymmetry of portfolio returns and a mean-risk-skewness model for portfolio selection will be proposed in uncertain environment. Here, the returns of the securities are regarded as uncertain variables which are estimated by experienced experts instead of historical data. Furthermore, the corresponding variations and crisp forms of the model are considered. To solve the proposed optimization models, a hybrid intelligent algorithm is designed. Finally, the feasibility and necessity of the hybrid intelligent algorithm and the application of the proposed models are illustrated by two numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we propose an arc-search infeasible-interior-point method based on the wide neighbourhoodfor linear complementarity problems over symmetric cones with the Cartesian-property (-SCLCP). The algorithm searches the optimizers along the ellipses that approximate the central path. Moreover, we derive the complexity bound of the algorithm which coincides with the currently best known theoretical complexity bounds for the short step path-following algorithm. Some numerical results are provided to demonstrate the computational performance of the proposed algorithm. [ABSTRACT FROM PUBLISHER]

Two projection algorithms for solving the SFP by combining the projection method for the equilibrium problem are introduced and the gradient method for a considered inclusion is derived. The proposed algorithm includes the simultaneous iterative algorithm as special case which has been proposed by Moudafi and Al-Shemas for solving the split equality common fixed-point problem. The proposed algorithms are based on inertial subgradient extragradient method with line-search process, hybrid steepest-descent method, and viscosity approximation method. [Extracted from the article]

In this paper, we investigate the properties of consistent conjectural variations equilibrium developed for a single-commodity oligopoly. Although, in general, the consistent conjectures are distinct from those of Cournot-Nash, we establish the following remarkable fact. Define a meta-game as such where the players are the same agents as in the original oligopoly but now using the conjectures as their strategies. Then the consistent conjectures of the original oligopoly game provide for the Cournot-Nash optimal strategies in the meta-game. [ABSTRACT FROM AUTHOR]

This paper deals with COTS evaluation and selection for developing a modular software system under single application development task. We consider both quantitative and qualitative criteria which fulfils the specific needs of a software system. We use analytical hierarchy process (AHP) technique for evaluating the fitness of COTS components based upon various criteria and sub-criteria thereby providing overall score of each COTS component. We develop optimization models integrating AHP and multi-criteria decision-making, which aim at: (i) maximize the total value of purchasing (TVP) subject to budget, compatibility and reliability constraints, and (ii) maximize TVP and minimize the total cost of purchase simultaneously subject to compatibility and reliability constraints. The efficiency of the models is illustrated by means of numerical illustrations. [ABSTRACT FROM PUBLISHER]

In this paper, we establish some results which exhibit an application of convexificators in vector optimization problems (VOPs) and vector variational inequaities involving locally Lipschitz functions. We formulate vector variational inequalities of Stampacchia and Minty type in terms of convexificators and use these vector variational inequalities as a tool to find out necessary and sufficient conditions for a point to be a vector minimal point of the VOP. We also consider the corresponding weak versions of the vector variational inequalities and establish several results to find out weak vector minimal points. [ABSTRACT FROM PUBLISHER]

In this paper, we study the generalized Hadamard well-posedness of infinite vector optimization problems (IVOP). Without the assumption of continuity with respect to the first variable, the upper semicontinuity and closedness of constraint set mappings are established. Under weaker assumptions, sufficient conditions of generalized Hadamard well-posedness for IVOP are obtained under perturbations of both the objective function and the constraint set. We apply our results to the semi-infinite vector optimization problem and the semi-infinite multi-objective optimization problem. [ABSTRACT FROM AUTHOR]

In this paper, by virtue of the epigraph technique, we first introduce some new regularity conditions and then obtain some complete characterizations of the Fenchel–Lagrange duality and the stable Fenchel–Lagrange duality for a new class of DC optimization involving a composite function. Moreover, we apply the strong and stable strong duality results to obtain some extended (stable) Farkas lemmas and (stable) alternative type theorems for this DC optimization problem. As applications, we obtain the corresponding results for a composed convex optimization problem, a DC optimization problem, and a convex optimization problem with a linear operator, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we present a nonsmooth trust region method for solving linearly constrained optimization problems with a locally Lipschitz objective function. Using the approximation of the steepest descent direction, a quadratic approximation of the objective function is constructed. The null space technique is applied to handle the constraints of the quadratic subproblem. Next, the CG-Steihaug method is applied to solve the new approximation quadratic model with only the trust region constraint. Finally, the convergence of presented algorithm is proved. This algorithm is implemented in the MATLAB environment and the numerical results are reported. [ABSTRACT FROM AUTHOR]

In this paper, some proximal point algorithms (PPAs) for maximal monotone operators in Banach spaces are considered. We obtain some results on the boundedness and the convergence of sequences generated by the PPAs with some assumptions. We can also get zero of maximal-expansive operator and maximal-monotone at zero operator using this method and then we apply it for finding a solution of the equilibrium problem. [ABSTRACT FROM AUTHOR]

In this paper, we consider multiobjective semi-infinite optimization problems which are defined in a finite-dimensional space by finitely many objective functions and infinitely many inequality constraints. We present duality results both for the convex and nonconvex case. In particular, we show weak, strong and converse duality with respect to both efficiency and weak efficiency. Moreover, the property of being a locally properly efficient point plays a crucial role in the nonconvex case. [ABSTRACT FROM AUTHOR]

This paper presents a numerical scheme for solving fractional optimal control. The fractional derivative in this problem is in the Riemann–Liouville sense. The proposed method, based upon the method of moments, converts the fractional optimal control problem to a semidefinite optimization problem; namely, the nonlinear optimal control problem is converted to a convex optimization problem. The Grunwald–Letnikov formula is also used as an approximation for fractional derivative. The solution of fractional optimal control problem is found by solving the semidefinite optimization problem. Finally, numerical examples are presented to show the performance of the method. [ABSTRACT FROM AUTHOR]

In this paper, we generalize Darvay’s technique for linear programming tohorizontal linear complementarity problem over the Cartesian product of symmetric cones, or briefly Cartesian-SCHLCP, using Euclidean Jordan algebras. This problem is a comprehensive optimization problem over symmetric cones, so that handling this problem removes the necessity of dealing with many other ones. This paper is among the first ones which study interior point methods (IPMs) for the generic Cartesian-SCHLCP problems. The symmetrization of the search directions used in this paper is based on the Nesterov–Todd (NT) scaling scheme, and only full-NT steps are used at each iteration. So, line searches are not longer needed. The derived complexity bound matches the best obtained one for infeasible IPMs with small updates. [ABSTRACT FROM AUTHOR]

This paper studies multiobjective optimal control problems in presence of constraints in the discrete time framework. Both the finite- and infinite-horizon settings are considered. The paper provides necessary conditions of Pareto optimality under lighter smoothness assumptions compared to the previously obtained results. These conditions are given in the form of weak and strong Pontryagin principles which generalize the existing ones. To obtain some of these results, we provide new multiplier rules for multiobjective static optimization problems and new Pontryagin principles for the finite horizon multiobjective optimal control problems. [ABSTRACT FROM AUTHOR]

In this paper, first, we show that the following optimization problem always has an optimal solution which does not depend on the matrix norm, and give a sufficient condition for the uniqueness of optimal solutions:(Section.Display) whereis any unitarily invariant norm,, andor. Then, we characterize the optimal solutions and also give sufficient conditions for the uniqueness and the rank-one property of optimal solutions for the following optimization problem:(Section.Display) where, is the inner product of matrices;,,, and’s,are different arbitrary norms. [ABSTRACT FROM AUTHOR]

In this paper, the local weak Pareto optimality conditions for nonsmooth vector optimization problems are presented on Hadamard manifolds. Some convergence results of the parallel proximal point algorithms for finding a solution of vector optimization problems are established. [ABSTRACT FROM AUTHOR]

In this paper, we present a wide neighbourhood primal-dual interior-point algorithm for semidefinite optimization. We define a new wide neighbourhood W (β , τ) , which is an extension of the Darvay-Takács's wide neighbourhood for linear optimization to semidefinite optimization. We use an algebraic reformulation of the centering equation along with Zhang's symmetrization operator to create symmetric search directions. After applying Newton's method to the resulting system, we recommend using the square root function for simplicity. We consider the Newton direction as the sum of two other directions, corresponding to the negative and positive parts of the right-hand side of the centring equation. Starting with a feasible point (X 0 , y 0 , S 0) ∈ W (β , τ) , we prove that our algorithm has complexity the same as small neighbourhood path-following algorithms for semidefinite optimization that use the Nesterov-Todd directions. To the best of our knowledge, this is the first large neighbourhood path-following interior-point algorithm that uses a different neighbourhood from those available for semidefinite optimization. [ABSTRACT FROM AUTHOR]

In this paper, we first discuss the Painlevé–Kuratowski set convergence of (weak) minimal point set for a convex set, when the set and the ordering cone are both perturbed. Next, we consider a convex vector optimization problem, and take into account perturbations with respect to the feasible set, the objective function and the ordering cone. For this problem, by assuming that the data of the approximate problems converge to the data of the original problem in the sense of Painlevé–Kuratowski convergence and continuous convergence, we establish the Painlevé–Kuratowski set convergence of (weak) minimal point and (weak) efficient point sets of the approximate problems to the corresponding ones of original problem. We also compare our main theorems with existing results related to the same topic. [ABSTRACT FROM PUBLISHER]

Direct-search algorithms form one of the main classes of algorithms for smooth unconstrained derivative-free optimization, due to their simplicity and their well-established convergence results. They proceed by iteratively looking for improvement along some vectors or directions. In the presence of smoothness, first-order global convergence comes from the ability of the vectors to approximate the steepest descent direction, which can be quantified by a first-order criticality (cosine) measure. The use of a set of vectors with a positive cosine measure together with the imposition of a sufficient decrease condition to accept new iterates leads to a convergence result as well as a worst-case complexity bound. In this paper, we present a second-order study of a general class of direct-search methods. We start by proving a weak second-order convergence result related to a criticality measure defined along the directions used throughout the iterations. Extensions of this result to obtain a true second-order optimality one are discussed, one possibility being a method using approximate Hessian eigenvectors as directions (which is proved to be truly second-order globally convergent). Numerically guaranteeing such a convergence can be rather expensive to ensure, as it is indicated by the worst-case complexity analysis provided in this paper, but turns out to be appropriate for some pathological examples. [ABSTRACT FROM PUBLISHER]

The analysis of the principal-agent problem usually requires the classical first-order approach (FOA). However, the validity of the FOA makes restrictive assumptions on the problem under consideration such as the convexity of the distribution function condition. The main aim of this paper is to compute the optimal wages and characterize a closed form solution to the risk-neutral principal-agent problem with limited liability constraints. The development in this paper mainly invokes certain techniques in the semi-infinite programming rather than the FOA. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new inexact dual decomposition algorithm for solving separable convex optimization problems. This algorithm is a combination of three techniques: dual Lagrangian decomposition, smoothing and excessive gap. The algorithm has low computational complexity since it consists in only one primal step and two dual steps at each iteration and allows one to solve the subproblem of each component inexactly and in parallel. Moreover, the algorithmic parameters are updated automatically without any tuning strategy as it happens in augmented Lagrangian approaches. We analyse the convergence of the algorithm and estimate itsanalytical worst-case complexity for both the primal–dual suboptimality and the primal feasibility violation, whereis a given accuracy. Extensive numerical tests confirm that our method is numerically more efficient than the classical decomposition methods from the literature. [ABSTRACT FROM AUTHOR]

In this paper, a new nonlinear scalarization function, which is a generalization of the oriented distance function, is introduced. Some properties of the function are discussed. Then the function is applied to obtain some new optimality conditions and scalar representations for set-valued vector optimization problems with set optimization criteria. In terms of the function and the image space analysis, some new alternative results for generalized parametric systems are derived. [ABSTRACT FROM AUTHOR]

We study semi-convex frontier (SCF) optimization problems where objective functions can be semi-convex and constraint sets can be non-polyhedron, which stem from a growing range of optimization applications such as frontier analysis, multi-objective programming in economics. The new findings of this paper can be summarized as follows: (1) We characterize non-dominated points of a non-polyhedron optimal solution set of a semi-convex frontier program. (2) We obtain optimality conditions of a constant modulus SCF program, of which the objective function is semi-convex with a constant semiconvexity modulus. (3) We obtain a non-smooth Hölder stability of the optimal solutions of a semiconvex frontier program. (4) We use generalized differentiability to establish sensitivity analysis of the optimal value function of a semi-convex frontier program. [ABSTRACT FROM AUTHOR]

Concave gauge functions were introduced to give an analytical representation of cones. In particular, they give a simple and a practical representation of the positive orthant. There is a wide choice of concave gauge functions with interesting properties, representing the same cone. Besides the fact that a concave gauge cannot be identically zero on a cone(), it may be continuous, differentiable and evenon its interior. The purpose of the present paper is to present another approach to penalizing the positivity constraints of a linear programme using an arbitrary strictly quasi-concave gauge representation. Throughout the paper, we generalize the concept of the central path and the analytic centre in terms of these gauges, introduce the differential barrier concept and establish its relationship with strict quasi-concavity. [ABSTRACT FROM AUTHOR]

Inverse data envelopment analysis (InDEA) is a well-known approach for short-term forecasting of a given decision-making unit (DMU). The conventional InDEA models use the production possibility set (PPS) that is composed of an evaluated DMU with current inputs and outputs. In this paper, we replace the fluctuated DMU with a modified DMU involving renewal inputs and outputs in the PPS since the DMU with current data cannot be allowed to establish the new PPS. Besides, the classical DEA models such as InDEA are assumed to consider perfect knowledge of the input and output values but in numerous situations, this assumption may not be realistic. The observed values of the data in these situations can sometimes be defined as interval numbers instead of crisp numbers. Here, we extend the InDEA model to interval data for evaluating the relative efficiency of DMUs. The proposed models determine the lower and upper bounds of the inputs of a given DMU separately when its interval outputs are changed in the performance analysis process. We aim to remain the current interval efficiency of a considered DMU and the interval efficiencies of the remaining DMUs fixed or even improve compared with the current interval efficiencies. [ABSTRACT FROM AUTHOR]

In this paper, existence of critical point and weak efficient point of vector optimization problem is studied. A sequence of points in n-dimension is generated using positive definite matrices like Quasi-Newton method. It is proved that accumulation points of this sequence are critical points or weak efficient points under different conditions. An algorithm is provided in this context. This method is free from any kind of priori chosen weighting factors or any other form of a priori ranking or ordering information for objective functions. Also, this method does not depend upon initial point. The algorithm is verified in numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we establish new necessary conditions for-minimizers of vector optimization problems with and without geometric constraints with respect tononconvex and nonsolid ordering sets. We use the dual-space approach based on advanced tools of variational analysis and generalized differentiation. It allows us to handle vector optimization problems without converting them to equivalent problems of scalar optimization. They are new even in vector optimization for ordering cones. [ABSTRACT FROM AUTHOR]