Semiparametric proper efficiency principles and duality models for constrained multiobjective fractional optimal control problems containing arbitrary norms

Semiparametric necessary and sufficient proper efficiency conditions are established for a class of constrained multiobjective fractional optimal control problems with linear dynamics, containing arbitrary norms. Moreover, utilizing these proper efficiency results, eight semiparametric duality models are formulated and appropriate duality theorems are proved. These proper efficiency and duality criteria contain, as special cases, similar results for several classes of unorthodox optimal control problems with multiple, fractional, and conventional objective functions, which are particular cases of the main problem considered in this paper.     

Proper efficiency and duality for a class of multiobjective fractional variational problems containing arbitrary norms

Necessary and sufficient conditions are established for properly efficient solutions of a class of nonsmooth nonconvex variational problems with multiple fractional objective functions and nonlinear inequality constraints. Based on these proper efficiency criteria. two multiobjective dual problems are constructed and appropriate duality theorems are proved. These proper efficiency and duality results also contain as special cases similar rcsults fer constrained variational problems with multiplei fractional. and conventional objective functions, which are particular cases of the main variational problem considered in this paper     

Optimality conditions and duality models for generalized fractional programming problems containing locally subdifferentiable and ρ:-convex functions

Both parametric and nonparametric necessary and sufficient optimality conditions are established for a class of nonsmooth generalized fractional programming problems containing ρ-convex functions. Subsequently, these optimality criteria are utilized as a basis for constructing two parametric and four parameter-free duality models and proving appropriate duality theorems. Several classes of generalized fractional programming problems, including those with arbitrary norms, square roots of positive semidefinite quadratic forms, support functions, continuous max functions, and discrete max functions, which can be viewed as special cases of the main problem are briefly discussed. The optimality and duality results developed here also contain, as special cases, similar results for nonsmooth problems with fractional, discrete max, and conventional objective functions which are particular cases of the main problem considered in this paper     

Duality for minmax programming involving V-invex functions

Sufficient optimality conditions and duality results for a class of minmax programming problems are obtained under V-invexity type assumptions on objective and constraint functions. Applications of these results to certain fractional and generalized fractional programming problems are also presented     

Multiple objective fractional programming involving semilocally type I-preinvex and related functions

Sufficient optimality conditions are obtained for a nonlinear multiple objective fractional programming problem involving η-semidifferentiable type I-preinvex and related functions. Furthermore, a general dual is formulated and duality results are proved under the assumptions of generalized semilocally type I-preinvex and related functions. Our result generalize the results of Preda [V. Preda, Optimality and duality in fractional multiple objective programming involving semilocally preinvex and related functions, J. Math. Anal. Appl. 288 (2003) 365–382] and Stancu-Minasian [I.M. Stancu-Minasian, Optimality and duality in fractional programming involving semilocally preinvex and related functions, J. Inform. Optim. Sci. 23 (2002) 185–201].     

OPTIMALITY CONDITIONS AND DUALITY MODELS FOR A CLASS OF NONSMOOTH CONTINUOUS-TIME GENERALIZED FRACTIONAL PROGRAMMING PROBLEMS

Abstract

Both parametric and parameter-free stationary-point-type and saddle-point-type necessary and sufficient optimality conditions are established for a class of nonsmooth continuous-time generalized fractional programming problems with Volterra-type integral inequality and nonnegativity constraints. These optimality criteria are then utilized for constructing ten parametric and parameter-free Wolfe-type and Lagrangian-type dual problems and for proving weak, strong, and strict converse duality theorems. Furthermore, it is briefly pointed out how similar optimality and duality results can be obtained for two important special cases of the main problem containing arbitrary norms and square roots of positive semidefinite quadratic forms. All the results developed here are also applicable to continuous-time programming problems with fractional, discrete max, and conventional objective functions, which are special cases of the main problem studied in this paper.     

A modified objective function method for solving nonlinear multiobjective fractional programming problems

A new method is used for solving nonlinear multiobjective fractional programming problems having V-invex objective and constraint functions with respect to the same function η. In this approach, an equivalent vector programming problem is constructed by a modification of the objective fractional function in the original nonlinear multiobjective fractional problem. Furthermore, a modified Lagrange function is introduced for a constructed vector optimization problem. By the help of the modified Lagrange function, saddle point results are presented for the original nonlinear fractional programming problem with several ratios. Finally, a Mond-Weir type dual is associated, and weak, strong and converse duality results are established by using the introduced method with a modified function. To obtain these duality results between the original multiobjective fractional programming problem and its original Mond-Weir duals, a modified Mond-Weir vector dual problem with a modified objective function is constructed.     

Optimality and Duality for a Class of Nondifferentiable Multiobjective Fractional Programming Problems

In this paper, we consider a class of nondifferentiable multiobjective fractional programs in which each component of the objective function contains a term involving the support function of a compact convex set. We establish necessary and sufficient optimality conditions and duality results for weakly efficient solutions of nondifferentiable multiobjective fractional programming problems. This work was supported by Grant R01-2003-000-10825-0 from the Basic Research Program of KOSEF.     

Some Farkas-type results for fractional programming problems with DC functions

For a kind of fractional programming problem that the objective functions are the ratio of two DC (difference of convex) functions with finitely many convex constraints, in this paper, its dual problems are constructed, weak and strong duality assertions are given, and some sufficient and necessary optimality conditions which characterize their optimal solutions are obtained. Some recently obtained Farkas-type results for fractional programming problems that the objective functions are the ratio of a convex function to a concave function with finitely many convex constraints are the special cases of the general results of this paper.     

Symmetric duality for a class of nondifferentiable multi-objective fractional variational problems

We introduce a symmetric dual pair for a class of nondifferentiable multi-objective fractional variational problems. Weak, strong, converse and self duality relations are established under certain invexity assumptions. The paper includes extensions of previous symmetric duality results for multi-objective fractional variational problems obtained by Kim, Lee and Schaible [D.S. Kim, W.J. Lee, S. Schaible, Symmetric duality for invex multiobjective fractional variational problems, J. Math. Anal. Appl. 289 (2004) 505-521] and symmetric duality results for the static case obtained by Yang, Wang and Deng [X.M. Yang, S.Y. Wang, X.T. Deng, Symmetric duality for a class of multiobjective fractional programming problems, J. Math. Anal. Appl. 274 (2002) 279-295] to the dynamic case.     

Optimality and duality in fractional multiple objective programming involving semilocally preinvex and related functions

Necessary and sufficient optimality conditions are obtained for a nonlinear fractional multiple objective programming problem involving η-semidifferentiable functions. Also, a general dual is formulated and duality results are proved using concepts of generalized semilocally preinvex functions.     

On nonlinear multiple objective fractional programming involving semilocally type-I univex functions

The aim of this paper is to obtain sufficient optimality conditions for a nonlinear multiple objective fractional programming problem involving semilocally type-I univex and related functions. Furthermore, a general dual is formulated and duality results are proved under the assumptions of generalized semilocally type-I univex and related functions. Our results generalize several known results in the literature.     

在pseudo-invexity条件下的一个分式规划问题的对偶性

给出了一个不可微多目标分式变分问题,并利用有效性和真有效性概念,证明了在pseudo-invexity条件下与分式规划问题相关的弱对偶定理、强对偶定理及逆对偶定理.     

Efficiency Conditions and Duality for a Class of Multiobjective Fractional Programming Problems

A class of constrained multiobjective fractional programming problems is considered from a viewpoint of the generalized convexity. Some basic concepts about the generalized convexity of functions, including a unified formulation of generalized convexity, are presented. Based upon the concept of the generalized convexity, efficiency conditions and duality for a class of multiobjective fractional programming problems are obtained. For three types of duals of the multiobjective fractional programming problem, the corresponding duality theorems are also established.     

Higher order duality for a new class of nonconvex semi-infinite multiobjective fractional programming with support functions

In the paper, a new class of semi-infinite multiobjective fractional programming problems with support functions in the objective and constraint functions is considered. For such vector optimization problems, higher order dual problems in the sense of Mond-Weir and Schaible are defined. Then, various duality results between the considered multiobjective fractional semi-infinite programming problem and its higher order dual problems mentioned above are established under assumptions that the involved functions are higher order $\left(\Phi,\rho,\sigma^{\alpha}\right)$-type I functions. The results established in the paper generalize several similar results previously established in the literature.     

On duality in multiobjective semi-infinite optimization

Abstract

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.     

Conjugate maps and duality in multiobjective optimization

This paper considers duality in convex vector optimization. A vector optimization problem requires one to find all the efficient points of the attainable value set for given multiple objective functions. Embedding the primal problem into a family of perturbed problems enables one to define a dual problem in terms of the conjugate map of the perturbed objective function. Every solution of the stable primal problem is associated with a certain solution of the dual problem, which is characterized as a subgradient of the perturbed efficient value map. This pair of solutions also provides a saddle point of the Lagrangian map.     

B‐spline spectral method for constrained fractional optimal control problems

In this paper, we present a direct B‐spline spectral collocation method to approximate the solutions of fractional optimal control problems with inequality constraints. We use the location of the maximum of B‐spline functions as collocation points, which leads to sparse and nonsingular matrix B whose entries are the values of B‐spline functions at the collocation points. In this method, both the control and Caputo fractional derivative of the state are approximated by B‐spline functions. The fractional integral of these functions is computed by the Cox‐de Boor recursion formula. The convergence of the method is investigated. Several numerical examples are considered to indicate the efficiency of the method.     

Duality for multiobjective optimization problems with convex objective functions and D.C. constraints

In this paper we provide a duality theory for multiobjective optimization problems with convex objective functions and finitely many D.C. constraints. In order to do this, we study first the duality for a scalar convex optimization problem with inequality constraints defined by extended real-valued convex functions. For a family of multiobjective problems associated to the initial one we determine then, by means of the scalar duality results, their multiobjective dual problems. Finally, we consider as a special case the duality for the convex multiobjective optimization problem with convex constraints.     

Nonsmooth minimax fractional programming involving η-pseudolinear functions

This article deals with the necessary and sufficient optimality conditions for a class of nonsmooth minimax fractional programming problems with locally Lipschitz η-pseudolinear functions. Utilizing these optimality criteria, we formulate two types of dual models and establish weak and strong duality results. The results of this article extend several known results from the literature to a wider class of optimization problems.