11th International Conference on Stochastic Programming

In this article we establish necessary conditions for local Pareto and weak minima of multiobjective programming problems involving inequality, equality and set constraints in Banach spaces in terms of convexificators.     

On second-order optimality conditions for continuously Fréchet differentiable vector optimization problems

ABSTRACT

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.     

Selberg’s one-dimensional sieve with Buchstab weights of new type

In this paper we study Selberg's sieve method with Buchstab weights of new type. The theorem proved in this paper gives a more advantageous choice of the parameters of a one-dimensional weighted sieve as compared to previous results. Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 38–49, July, 1999.     

The improved screening curve method regarding existing units

In this study, a new method is developed to determine the least cost capacity expansion of a power system by using the screening curve method. The proposed methodology differs from the previous studies by its geometrical solution process to evaluate a capacity expansion problem considering both existing and candidate power plants. The algorithms are computationally more efficient and simple than the ones in previous studies for the same improvement. Further, the interpretation of the optimal capacity expansion plan is enhanced by explicitly exhibiting the results of all considered capacity expansion alternatives. The solution process can be interpreted as minimizing the long run marginal cost of supplying 1 megawatt of capacity during the whole year by finding the optimal combination of units. The developed method calculates and finds the cost polygon with the minimum area by moving along the intersection points of the screening curves to form trapezoids and then joining them to form cost polygons. The intersection points, which are needed to calculate the areas of the cost polygons, are found by using the Karush–Kuhn–Tucker conditions in a recursive manner. The last unit in the merit order of dispatching is determined by scenarios to yield an optimal capacity expansion plan. The scenarios are primarily based on a tradeoff between incurring investment costs by commissioning candidate units or taking online existing units with relatively higher variable costs compared to the candidate units.     

An (α ,β)-Optimal solution concept in Fuzzy Optimization problems

We propose an (α,β)-optimal solution concept of fuzzy optimization problem based on the possibility and necessity measures. It is well known that the set of all fuzzy numbers can be embedded into a Banach space isometrically and isomorphically. Inspired by this embedding theorem, we can transform the fuzzy optimization problem into a biobjective programming problem by applying the embedding function to the original fuzzy optimization problem. Then the (α,β)-optimal solutions of fuzzy optimization problem can be obtained by solving its corresponding biobjective programming problem. We also consider the fuzzy optimization problem with fuzzy coefficients (i.e., the coefficients are assumed as fuzzy numbers). Under a setting of core value of fuzzy numbers, we provide the Karush–Kuhn–Tucker optimality conditions and show that the optimal solution of its corresponding crisp optimization problem (the usual optimization problem) is also a (1,1)-optimal solution of the original fuzzy optimization problem.     

一类凸二次规划的新算法

本文给出了求解一类凸二次规划问题的新算法.这种算法既保留了传统算法的优点,又避免了其它算法中出现的添加人工变量过多、循环等问题.算例表明,这种算法是简便而有效的.     

Kuhn–Tucker Conditions for a Convex Programming Problem in Banach Spaces Partially Ordered by Cone with Empty Interior

The existence of saddle point of the Lagrange function for a convex programming problem in Banach spaces ordered by a cone with empty interior is established under a strong simultaneity condition. As a consequence, the Kuhn–Tucker conditions are derived. It is shown that the Slater and the strong simultaneity condition are equivalent if the cone determining the partial order has an interior.