In this paper, we develop optimality conditions and propose an algorithm for generalized fractional programming problems whose objective function is the maximum of finite ratios of difference of convex (dc) functions,...
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In this paper, we develop optimality conditions and propose an algorithm for generalized fractional programming problems whose objective function is the maximum of finite ratios of difference of convex (dc) functions, with dc constraints, that we will call later, DC-GFP. Such problems are generally nonsmooth and nonconvex. We first give in this work, optimality conditions for such problems, by means of convex analysis tools. For solving DC-GFP, the use of dinkelbach-type algorithms conducts to nonconvex subproblems, which causes the failure of the latter since it requires finding a global minimum for these subprograms. To overcome this difficulty, we propose a DC-dinkelbach-type algorithm in which we overestimate the objective function in these subproblems by a convex function, and the constraints set by an inner convex subset of the latter, which leads to convex subproblems. We show that every cluster point of the sequence of optimal solutions of these subproblems satisfies necessary optimality conditions of KKT type. Finally we end with some numerical tests to illustrate the behavior of our algorithm.
This paper deals with minimax fractional programs whose objective functions are the maximum of finite ratios of convex functions, with arbitrary convex constraints set. For such problems, dinkelbach-type algorithms fa...
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This paper deals with minimax fractional programs whose objective functions are the maximum of finite ratios of convex functions, with arbitrary convex constraints set. For such problems, dinkelbach-type algorithms fail to work since the parametric subproblems may be nonconvex, whereas the latter need a global optimal solution of these subproblems. We give necessary optimality conditions for such problems, by means of convex analysis tools. We then propose a method, based on solving approximately a sequence of parametric convex problems, which acts as dc (difference of convex functions) algorithm, if the parameter is positive and as dinkelbach algorithm if not. We show that every cluster point of the sequence of optimal solutions of these subproblems satisfies necessary optimality conditions of KKT criticality type, that are also of Clarke stationarity type. Finally we end with some numerical tests to illustrate the behaviour of the algorithm.
In this paper, we present several new implementable methods for solving a generalized fractional program with convex data. They are dinkelbach-type methods where a prox-regularization term is added to avoid the numeri...
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In this paper, we present several new implementable methods for solving a generalized fractional program with convex data. They are dinkelbach-type methods where a prox-regularization term is added to avoid the numerical difficulties arising when the solution of the problem is not unique. In these methods, at each iteration a regularized parametric problem is solved inexactly to obtain an approximation of the optimal value of the problem. Since the parametric problem is nonsmooth and convex, we propose to solve it by using a classical bundle method where the parameter is updated after each 'serious step'. We mainly study two kinds of such steps, and we prove the convergence and the rate of convergence of each of the corresponding methods. Finally, we present some numerical experience to illustrate the behavior of the proposed algorithms, and we discuss the practical efficiency of each one.
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