To calculate the closed cone induced by epigraph of conjugate functions of sum of linear mappings and matrix norms, we give the formula of the subdifferential of matrix norm ∥ · ∥1 at 0. We consider a sublinear matrix optimization problem (P) involving a matrix norm ∥ · ∥1. and then we show that the existence of optimal solutions for (P) is closely related to its zero solution. Moreover we consider a convex matrix optimization problem (CP) involving matrix norm ∥ · ∥1, and establish an optimality theorem (CP) which holds without any constraint qualification and expressed with the subdifferential. We give an example illustrating the optimality theorem.

We consider a linear fractional optimization problem (FP) involving integral function defined on C^n [0, 1], and then characterize solution sets for the problem (FP) in terms of sequential Lagrange multipliers of a known solution of (FP). Moreover, we give an example illustrating our characterization of solution set.

We consider a linear fractional optimization problem involving integral functions defined on C-n[0, 1] and obtain an optimality theorem for the problem which holds without any constraint qualification. We give an example to demonstrate how to use the optimality theorem for finding the optimal solutions.

We consider a scmidcfinite linear fractional vector optimization problem (FVP) and establish optimality theorems for weakly efficient solutions for (FVP), which hold without any constraint qualification. We first discuss the relation between weakly efficient solution of (FVP) and one of its related linear vector optimization problem (LVP). By using the relation and the maximum function of objetive functions of (FVP), we obtain our optimality theorems for weakly efficient solutions for (FVP), and then we give examples showing how to use our optimality theorems for finding weakly efficient solutions for (FVP). Moreover, we formulate vector dual problem (VD) for (FVP), which is a kind of vector version of Wolfe dual problem, and establish duality theorems for (FVP) and (VD), which hold without any constraint qualification.

We consider a linear fractional optimization problem (LFOP) defined on an Euclidean Jordan algebras. We obtain an optimality theorem for the LFOP, which holds without any constraint qualification. Moreover, we formulate the non-fractional dual problem of the LFOP and then prove the duality theorems (weak duality theorem and strong duality theorem), which hold without any constraint qualification. Furthermore, we characterize the solution set of the LFOP by using the optimality conditions. We also discuss methodologies for the LFOP.

We study epsilon-solutions for a semidefinite linear fractional optimiza-tion problem (SLF). We obtain sequential optimality theorems for epsilon-solutions for (SLF), which are expressed with sequences and hold without any constraint qual-ification. Moreover, we formulate the non-fractional dual problem of (SLF) and then prove the sequential duality theorems (epsilon-weak duality theorem and epsilon-strong duality theorem), which holds without any constraint qualification.

Recently, semidefinite optimization problems have been intensively studied since many optimization problem can be transformed into the problems and the problems are computationally tractable. In this paper, we consider a semidefinite linear fractional optimization problem (SLF), and obtain optimality conditions for (SLF) which hold without any constraint qualification and which are expressed by sequences. By using the optimality conditions, we formulate the nonfractional dual problem (D) for (SLF), which is expressed by sequences, and prove the weak duality theorem and the strong duality theorem which hold between (SLF) and (D). The strong duality theorem holds without any constraint qualification. Furthermore, we characterize the solution set of (SLF) by using the optimality conditions.

We characterize the solution set for a second-order cone linear fractional optimization problem (P). We present sequential Lagrange multiplier characterizations of the solution set for the problem (P) in terms of sequential Lagrange multipliers of a known solution of (P).

In this paper, we consider a nondifferetiable fractional optimization problem (GFP) for locally Lipschitz functions and its Mond-Weir dual problem (DGFP), and then we define (V, p)-invexity conditions for vector-valued functions. By using (strict) (V, p)-invexity conditions for involving functions in (GFP) and the Kuhn-Tucker necessary optimality theorem for (GFP), we obtain the strict converse duality theorem for (GFP), which says that under (strict) (V, p)invexity conditions for functions and a constraint qualification for (GFP), the solution of (DGFP) is the solution of (GFP).

Recently, semidefinite optimization problems have been intensively studied since many optimization problem can be changed into the problems and the problems are very tractable. In this paper, we consider a semidefinite linear fractional optimization problem (SLF), and formulate a semidefinite linear optimization problem (SLD) as the dual problem of (SLF). We directly prove the weak duality theorem, the strong duality theorem and the converse duality theorem which hold between (SLF) and (SLD).

We establish necessary and sufficient optimality conditions for a class of generalized nondifferentiable fractional optimization programming problems. Moreover, we prove the weak and strong duality theorems under (V, ${\rho}$)-invexity assumption.

In this paper, we prove a sufficient optimality theorems for the problem(FP) under(V, ${\rho}$)-invexity assumption. And we give Mond-Weir type dual problem and proved weak and strong duality theorem under (V, ${\rho}$)-invexity

In this paper, we consider a generalized fractional robust optimization problem (FP). Establishing a nonfractional optimization problem (NFP) equivalent to (FP), we establish necessary optimality conditions and duality results.

In this paper, we consider a fractional robust optimization problem (FP) and give necessary optimality theorems for (FP). Establishing a nonfractional optimization problem (NFP) equivalent to (FP), we formulate a Mond-Weir type dual problem for (FP) and prove duality theorems for (FP).

A multiobjective fractional optimization problem (MFP), which consists of more than two fractional objective functions with convex numerator functions and convex denominator functions, finitely many convex constraint functions, and a geometric constraint set, is considered. Using parametric approach, we transform the problem (MFP) into the non-fractional multiobjective convex optimization problem (NMCP) v with parametric v ∈ ℝ p , and then give the equivalent relation between (weakly) ε-efficient solution of (MFP) and (weakly) Open image in new window -efficient solution of Open image in new window . Using the equivalent relations, we obtain ε- optimality conditions for (weakly) ε- efficient solution for (MFP). Furthermore, we present examples illustrating the main results of this study.

A convex vector optimization problem, which consists of more than two convex objective functions and finitely many convex constraint functions, is considered. In this paper, we discuss epsilon-efficient solutions and weakly epsilon-efficient solutions for the convex vector optimization problem and obtain epsilon-optimality theorems for such solutions of which hold without any constraint conditions and are expressed by sequences. Moreover, we obtain epsilon-optimality theorems for the convex vector optimization problem which hold under certain constraint qualifications.

A generalized nondifferentiable fractional optimization problem (GFP), which consists of a maximum objective function defined by finite fractional functions with differentiable functions and support functions, and a constraint set defined by differentiable functions, is considered. Recently, Kim et al. [Journal of Optimization Theory and Applications 129 (2006), no. 1, 131-146] proved optimality theorems and duality theorems for a nondifferentiable multiobjective fractional programming problem (MFP), which consists of a vector-valued function whose components are fractional functions with differentiable functions and support functions, and a constraint set defined by differentiable functions. In fact if x is a solution of (GFP), then x is a weakly efficient solution of (MFP), but the converse may not be true. So, it seems to be not trivial that we apply the approach of Kim et al. to (GFP). However, modifying their approach, we obtain optimality conditions and duality results for (GFP).

In this paper, we discuss $\epsilon$-optimality conditions and $\epsilon$-saddle point theorems for $\epsilon$-approximate solutions for convex semidefinite optimization problem which hold under a weakened constraint qualification or which hold without any constraint qualification. Moreover, we formulate a Wolfe type dual problem for the convex semidefinite optimization problem, and prove $\epsilon$-weak duality and $\epsilon$-strong duality between the primal problem and the dual problem, which hold under a weakened constraint qualification.