Majid Soleimani-damaneh | University of Tehran (original) (raw)
Papers by Majid Soleimani-damaneh
arXiv: Optimization and Control, 2015
This paper is divided to two parts. In the first part, we provide elementary proofs for some impo... more This paper is divided to two parts. In the first part, we provide elementary proofs for some important results in multi-objective optimization. The given proofs are so simple and short in compared to the existing ones. Also, a Pareto reducibility result is extended from efficiency to proper efficiency. The second part is devoted to the connections between nonemptiness, RpgeqqR^p_{\geqq}Rpgeqq-(semi)compactness, external stability and connectedness of the set of nondominated solutions in multi-objective optimization. Furthermore, it is shown that some assumption in an important result, concerning connectedness, is redundant and should be removed.
Bulletin of The Iranian Mathematical Society, 2016
In this paper, we deal with the subdifferential concept on Hadamard spaces. Flat Hadamard spaces ... more In this paper, we deal with the subdifferential concept on Hadamard spaces. Flat Hadamard spaces are characterized, and neces- sary and sufficient conditions are presented to prove that the subdiffer- ential set in Hadamard spaces is nonempty. Proximal subdifferential in Hadamard spaces is addressed and some basic properties are highlighted. Finally, a density theorem for subdifferential set is established.
In very recent decades, Scale Elasticity (SE), as a quantitative characterization of Returns to S... more In very recent decades, Scale Elasticity (SE), as a quantitative characterization of Returns to Scale (RTS), has been an attractive research issue in convex DEA models. However, we show that the existing Scale Elasticity (SE) measure does not work properly under nonconvex FDH technologies. Due to this, we define an SE counterpart in FDH models. To this end, two new quantities, called maximum incremental and minimum decremental ratios are introduced; and a polynomial-time procedure is developed to calculate them. The second part of the paper is devoted to introducing and investigating left- and right-hand RTS notions in FDH models. Some necessary and sufficient conditions are established, leading to a polynomial-time test for identification of the one-sided RTS status of DMUs. Finally, the relationships between one-sided RTS, Global RTS (GRS), and the newly-defined ratios are established.
European Journal of Operational Research
RAIRO - Operations Research
Compromise solutions, as feasible points as close as possible to the ideal (utopia) point, are im... more Compromise solutions, as feasible points as close as possible to the ideal (utopia) point, are important solutions in multiple objective programming. It is known in the literature that each compromise solution is a properly efficient solution if the sum of the image set and conical ordering cone is closed. In this paper, we prove the same result in a general setting without any assumption.
Journal of Global Optimization
Journal of Optimization Theory and Applications
Optimization Methods and Software
ABSTRACT In this paper, we introduce the anchor point notion for multiobjective optimization prob... more ABSTRACT In this paper, we introduce the anchor point notion for multiobjective optimization problems. These points are special properly efficient points. We characterize anchor points via some theorems. The theoretical results lead to two tests which are able to recognize whether a given feasible point is an anchor point or not. One of these tests works by solving several linear programming problems and another one is done by solving a single mixed-integer programming problem. We compare the obtained procedures by implementing them on various randomly generated numerical examples. Some discussions about the density and connectedness of the set of nonanchor points are provided as well.
Optimization
Abstract The main aim of this paper is to investigate weakly/properly/robust efficient solutions ... more Abstract The main aim of this paper is to investigate weakly/properly/robust efficient solutions of a nonsmooth semi-infinite multiobjective programming problem, in terms of convexificators. In some of the results, we assume the feasible set to be locally star-shaped. The appearing functions are not necessarily smooth/locally Lipschitz/convex. First, constraint qualifications and the normal cone to the feasible set are studied. Then, as a major part of the paper, various necessary and sufficient optimality conditions for solutions of the problem under consideration are presented. The paper is closed by a linear approximation problem to detect the solutions and by studying a gap function.
ESAIM: Control, Optimisation and Calculus of Variations, 2015
This paper deals with the Quasi Variational Inequality (QVI) problem on Banach spaces. Necessary ... more This paper deals with the Quasi Variational Inequality (QVI) problem on Banach spaces. Necessary and sufficient conditions for the solutions of QVI are given, using the subdifferential of distance function and the normal cone. A dual problem corresponding to QVI is constructed and strong duality is established. The solutions of dual problem are characterized according to the saddle points of the Lagrangian map. A gap function for dual of QVI is presented and its properties are established. Moreover, some applied examples are addressed.
Journal of Convex Analysis
Convexity and generalized convexity play a central role in mathematical programming for duality r... more Convexity and generalized convexity play a central role in mathematical programming for duality results and in order to characterize the solutions set. In this paper, taking in mind Craven's notion of K-invexity function (when K is a cone in R-n) and Martin's notion of Karush-Kuhn-Tucker invexity (hereafter KKT-invexity), we define new notions of generalized convexity for a multiobjective problem with conic constraints. These new notions are both necessary and sufficient to ensure every Karush-Kuhn-Tucker point is a solution. The study of the solutions is also done through the solutions of an associated scalar problem. A Mond-Weir type dual problem is formulated and weak and strong duality results are provided. The notions and results that exist in the literature up to now are particular instances of the ones presented here.
International Journal of Computer Mathematics, 2011
ABSTRACT In this paper, we discuss modelling and solving some multiobjective optimization problem... more ABSTRACT In this paper, we discuss modelling and solving some multiobjective optimization problems arising in biology. A class of comparison problems for string selection in molecular biology and a relocation problem in conservation biology are modelled as multiobjective optimization programmes. Some discussions about applications, solvability and different variants of the obtained models are given, as well. A crucial part of the study is based upon the Pareto optimization which refers to the Pareto solutions of multiobjective optimization problems. For such solution, improvement of some objective function can only be obtained at the expense of the deterioration of at least one other objective function.
Mathematical Methods of Operations Research, 2017
In this paper, we consider a nonsmooth optimization problem with a convex feasible set described ... more In this paper, we consider a nonsmooth optimization problem with a convex feasible set described by constraint functions which are neither convex nor differentiable nor locally Lipschitz necessarily. Utilizing upper regular convexificators, we characterize the normal cone of the feasible set and derive KKT type necessary and sufficient optimality conditions. Under some assumptions, we show that the set of KKT multipliers is bounded. We also characterize the set of optimal solutions and introduce a linear approximation corresponding to the original problem which is useful in checking optimality. The obtained outcomes extend various results existing in the literature to a more general setting.
Nonlinear Analysis: Theory, Methods & Applications, 2007
This paper establishes some sufficient conditions for optimality and proper optimality for multip... more This paper establishes some sufficient conditions for optimality and proper optimality for multiple-objective programs in Banach spaces, after extending the concept of vector invexity.
Topology, 2009
As mentioned in the abundant literature of convex analysis, convexity plays a vital role in many ... more As mentioned in the abundant literature of convex analysis, convexity plays a vital role in many pure and applied mathematical problems; see, eg, [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16] and [17] and the references therein. In recent years, many ...
SIAM Journal on Control and Optimization, 2010
ABSTRACT In this paper, new results, which exhibit some new applications for Mordukhovich'... more ABSTRACT In this paper, new results, which exhibit some new applications for Mordukhovich's subdifferential in nonsmooth optimization and variational problems, are established. Nonsmooth (fractional) multiobjective optimization problems in special Banach spaces are studied, and some necessary and sufficient conditions for weak Pareto-optimality for these problems are introduced. Through this work, we introduce into nonsmooth optimization theory in Banach algebras a new class of mathematical programming problems, which generalizes the notion of smooth KT-$(p,r)$-invexity. Some optimality conditions regarding the generalized KT-$(p,r)$-invexity notion and Kuhn-Tucker points are provided. Also, we seek a connection between linear (semi-) infinite programming and nonlinear programming. Some sufficient conditions for (proper) optimality under invexity are provided. A nonsmooth variational problem corresponding to a considered multiobjective problem is defined and the relations between the provided variational problem and the considered optimization problem are studied. The final part of the paper is devoted to illustrating a penalization mechanism, using the distance function as a tool, to provide some conditions to the solutions of the nonsmooth variational inequality problems. All results of the paper have been established in the absence of gradient vectors, using the properties of Mordukhovich's subdifferential in Asplund spaces.
arXiv: Optimization and Control, 2015
This paper is divided to two parts. In the first part, we provide elementary proofs for some impo... more This paper is divided to two parts. In the first part, we provide elementary proofs for some important results in multi-objective optimization. The given proofs are so simple and short in compared to the existing ones. Also, a Pareto reducibility result is extended from efficiency to proper efficiency. The second part is devoted to the connections between nonemptiness, RpgeqqR^p_{\geqq}Rpgeqq-(semi)compactness, external stability and connectedness of the set of nondominated solutions in multi-objective optimization. Furthermore, it is shown that some assumption in an important result, concerning connectedness, is redundant and should be removed.
Bulletin of The Iranian Mathematical Society, 2016
In this paper, we deal with the subdifferential concept on Hadamard spaces. Flat Hadamard spaces ... more In this paper, we deal with the subdifferential concept on Hadamard spaces. Flat Hadamard spaces are characterized, and neces- sary and sufficient conditions are presented to prove that the subdiffer- ential set in Hadamard spaces is nonempty. Proximal subdifferential in Hadamard spaces is addressed and some basic properties are highlighted. Finally, a density theorem for subdifferential set is established.
In very recent decades, Scale Elasticity (SE), as a quantitative characterization of Returns to S... more In very recent decades, Scale Elasticity (SE), as a quantitative characterization of Returns to Scale (RTS), has been an attractive research issue in convex DEA models. However, we show that the existing Scale Elasticity (SE) measure does not work properly under nonconvex FDH technologies. Due to this, we define an SE counterpart in FDH models. To this end, two new quantities, called maximum incremental and minimum decremental ratios are introduced; and a polynomial-time procedure is developed to calculate them. The second part of the paper is devoted to introducing and investigating left- and right-hand RTS notions in FDH models. Some necessary and sufficient conditions are established, leading to a polynomial-time test for identification of the one-sided RTS status of DMUs. Finally, the relationships between one-sided RTS, Global RTS (GRS), and the newly-defined ratios are established.
European Journal of Operational Research
RAIRO - Operations Research
Compromise solutions, as feasible points as close as possible to the ideal (utopia) point, are im... more Compromise solutions, as feasible points as close as possible to the ideal (utopia) point, are important solutions in multiple objective programming. It is known in the literature that each compromise solution is a properly efficient solution if the sum of the image set and conical ordering cone is closed. In this paper, we prove the same result in a general setting without any assumption.
Journal of Global Optimization
Journal of Optimization Theory and Applications
Optimization Methods and Software
ABSTRACT In this paper, we introduce the anchor point notion for multiobjective optimization prob... more ABSTRACT In this paper, we introduce the anchor point notion for multiobjective optimization problems. These points are special properly efficient points. We characterize anchor points via some theorems. The theoretical results lead to two tests which are able to recognize whether a given feasible point is an anchor point or not. One of these tests works by solving several linear programming problems and another one is done by solving a single mixed-integer programming problem. We compare the obtained procedures by implementing them on various randomly generated numerical examples. Some discussions about the density and connectedness of the set of nonanchor points are provided as well.
Optimization
Abstract The main aim of this paper is to investigate weakly/properly/robust efficient solutions ... more Abstract The main aim of this paper is to investigate weakly/properly/robust efficient solutions of a nonsmooth semi-infinite multiobjective programming problem, in terms of convexificators. In some of the results, we assume the feasible set to be locally star-shaped. The appearing functions are not necessarily smooth/locally Lipschitz/convex. First, constraint qualifications and the normal cone to the feasible set are studied. Then, as a major part of the paper, various necessary and sufficient optimality conditions for solutions of the problem under consideration are presented. The paper is closed by a linear approximation problem to detect the solutions and by studying a gap function.
ESAIM: Control, Optimisation and Calculus of Variations, 2015
This paper deals with the Quasi Variational Inequality (QVI) problem on Banach spaces. Necessary ... more This paper deals with the Quasi Variational Inequality (QVI) problem on Banach spaces. Necessary and sufficient conditions for the solutions of QVI are given, using the subdifferential of distance function and the normal cone. A dual problem corresponding to QVI is constructed and strong duality is established. The solutions of dual problem are characterized according to the saddle points of the Lagrangian map. A gap function for dual of QVI is presented and its properties are established. Moreover, some applied examples are addressed.
Journal of Convex Analysis
Convexity and generalized convexity play a central role in mathematical programming for duality r... more Convexity and generalized convexity play a central role in mathematical programming for duality results and in order to characterize the solutions set. In this paper, taking in mind Craven's notion of K-invexity function (when K is a cone in R-n) and Martin's notion of Karush-Kuhn-Tucker invexity (hereafter KKT-invexity), we define new notions of generalized convexity for a multiobjective problem with conic constraints. These new notions are both necessary and sufficient to ensure every Karush-Kuhn-Tucker point is a solution. The study of the solutions is also done through the solutions of an associated scalar problem. A Mond-Weir type dual problem is formulated and weak and strong duality results are provided. The notions and results that exist in the literature up to now are particular instances of the ones presented here.
International Journal of Computer Mathematics, 2011
ABSTRACT In this paper, we discuss modelling and solving some multiobjective optimization problem... more ABSTRACT In this paper, we discuss modelling and solving some multiobjective optimization problems arising in biology. A class of comparison problems for string selection in molecular biology and a relocation problem in conservation biology are modelled as multiobjective optimization programmes. Some discussions about applications, solvability and different variants of the obtained models are given, as well. A crucial part of the study is based upon the Pareto optimization which refers to the Pareto solutions of multiobjective optimization problems. For such solution, improvement of some objective function can only be obtained at the expense of the deterioration of at least one other objective function.
Mathematical Methods of Operations Research, 2017
In this paper, we consider a nonsmooth optimization problem with a convex feasible set described ... more In this paper, we consider a nonsmooth optimization problem with a convex feasible set described by constraint functions which are neither convex nor differentiable nor locally Lipschitz necessarily. Utilizing upper regular convexificators, we characterize the normal cone of the feasible set and derive KKT type necessary and sufficient optimality conditions. Under some assumptions, we show that the set of KKT multipliers is bounded. We also characterize the set of optimal solutions and introduce a linear approximation corresponding to the original problem which is useful in checking optimality. The obtained outcomes extend various results existing in the literature to a more general setting.
Nonlinear Analysis: Theory, Methods & Applications, 2007
This paper establishes some sufficient conditions for optimality and proper optimality for multip... more This paper establishes some sufficient conditions for optimality and proper optimality for multiple-objective programs in Banach spaces, after extending the concept of vector invexity.
Topology, 2009
As mentioned in the abundant literature of convex analysis, convexity plays a vital role in many ... more As mentioned in the abundant literature of convex analysis, convexity plays a vital role in many pure and applied mathematical problems; see, eg, [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16] and [17] and the references therein. In recent years, many ...
SIAM Journal on Control and Optimization, 2010
ABSTRACT In this paper, new results, which exhibit some new applications for Mordukhovich'... more ABSTRACT In this paper, new results, which exhibit some new applications for Mordukhovich's subdifferential in nonsmooth optimization and variational problems, are established. Nonsmooth (fractional) multiobjective optimization problems in special Banach spaces are studied, and some necessary and sufficient conditions for weak Pareto-optimality for these problems are introduced. Through this work, we introduce into nonsmooth optimization theory in Banach algebras a new class of mathematical programming problems, which generalizes the notion of smooth KT-$(p,r)$-invexity. Some optimality conditions regarding the generalized KT-$(p,r)$-invexity notion and Kuhn-Tucker points are provided. Also, we seek a connection between linear (semi-) infinite programming and nonlinear programming. Some sufficient conditions for (proper) optimality under invexity are provided. A nonsmooth variational problem corresponding to a considered multiobjective problem is defined and the relations between the provided variational problem and the considered optimization problem are studied. The final part of the paper is devoted to illustrating a penalization mechanism, using the distance function as a tool, to provide some conditions to the solutions of the nonsmooth variational inequality problems. All results of the paper have been established in the absence of gradient vectors, using the properties of Mordukhovich's subdifferential in Asplund spaces.