共查询到20条相似文献,搜索用时 15 毫秒
1.
On duality theory in multiobjective programming 总被引:5,自引:0,他引:5
D. T. Luc 《Journal of Optimization Theory and Applications》1984,43(4):557-582
In this paper, we study different vector-valued Lagrangian functions and we develop a duality theory based upon these functions for nonlinear multiobjective programming problems. The saddle-point theorem and the duality theorem are derived for these problems under appropriate convexity assumptions. We also give some relationships between multiobjective optimizations and scalarized problems. A duality theory obtained by using the concept of vector-valued conjugate functions is discussed.The author is grateful to the reviewer for many valuable comments and helpful suggestions. 相似文献
2.
Geometric consideration of duality in vector optimization 总被引:1,自引:0,他引:1
H. Nakayama 《Journal of Optimization Theory and Applications》1984,44(4):625-655
Recently, duality in vector optimization has been attracting the interest of many researchers. In order to derive duality in vector optimization, it seems natural to introduce some vector-valued Lagrangian functions with matrix (or linear operator, in some cases) multipliers. This paper gives an insight into the geometry of vector-valued Lagrangian functions and duality in vector optimization. It is observed that supporting cones for convex sets play a key role, as well as supporting hyperplanes, traditionally used in single-objective optimization.The author would like to express his sincere gratitude to Prof. T. Tanino of Tohoku University and to some anonymous referees for their valuable comments. 相似文献
3.
Using the concept of vector-valued Lagrangian functions, we characterize a special class of solutions,D-solutions, of a multiobjective programming problem with set functions in which the domination structure is described by a closed convex coneD. Properties of two perturbation functions, primal map and dual map, are also studied. Results lead to a general duality theorem.The authors greatly appreciate helpful and valuable comments and suggestions received from the referee. 相似文献
4.
Radu Ioan Boţ Sorin-Mihai Grad Gert Wanka 《Mathematical Methods of Operations Research》2007,65(3):417-444
A general duality framework in convex multiobjective optimization is established using the scalarization with K-strongly increasing functions and the conjugate duality for composed convex cone-constrained optimization problems. Other
scalarizations used in the literature arise as particular cases and the general duality is specialized for some of them, namely
linear scalarization, maximum (-linear) scalarization, set scalarization, (semi)norm scalarization and quadratic scalarization.
相似文献
5.
Francisco Guerra-Vázquez 《Optimization》2017,66(8):1237-1249
AbstractIn 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. 相似文献
6.
We show that a simple and elegant method of Bismut [J. Math. Analysis Appl., 44 (1973), pp. 384–404] for applying conjugate duality to convex problems of Bolza adapts directly to problems of utility maximization with portfolio constraints in mathematical finance. This gives a straightforward construction of an associated dual problem together with Euler–Lagrange and transversality relations, which are then used to establish existence of optimal portfolios in terms of solutions of the dual problem. The approach is completely synthetic, and does not require the rather difficult a priori hypothesis of a fictitious complete market for unconstrained optimization, which has been the standard approach for synthesizing optimal portfolios in problems of utility maximization with trading constraints. It also complements a duality synthesis of Rogers [Lecture Notes in Mathematics, No. LNM-1814, Springer-Verlag, New York, 2003, pp. 95–131] and Klein and Rogers [Math. Finance, 17 (2007), pp. 225–247] for general problems of utility maximization with market imperfections. 相似文献
7.
S. Nobakhtian 《Journal of Global Optimization》2008,41(1):103-115
We consider nonsmooth multiobjective fractional programming problems with inequality and equality constraints. We establish
the necessary and sufficient optimality conditions under various generalized invexity assumptions. In addition, we formulate
a mixed dual problem corresponding to primal problem, and discuss weak, strong and strict converse duality theorems.
This research was partially supported by Project no. 850203 and Center of Excellence for Mathematics, University of Isfahan,
Iran. 相似文献
8.
Guneet Bhatia 《Journal of Mathematical Analysis and Applications》2008,342(1):135-145
In this paper, higher order strong convexity for Lipschitz functions is introduced and is utilized to derive the optimality conditions for the new concept of strict minimizer of higher order for a multiobjective optimization problem. Variational inequality problem is introduced and its solutions are related to the strict minimizers of higher order for a multiobjective optimization problem. The notion of vector valued partial Lagrangian is also introduced and equivalence of the mixed saddle points of higher order and higher order minima are provided. 相似文献
9.
Necessary and sufficient conditions of Fritz John type for Pareto optimality of multiobjective programming problems are derived. This article suggests to establish a Wolfe-type duality theorem for nonlinear, nondifferentiable, convex multiobjective minimization problems. The vector Lagrangian and the generalized saddle point for Pareto optimality are studied. Some previously known results are shown to be special cases of the results described in this paper.This research was partly supported by the National Science Council, Taipei, ROC.The authors would like to thank the two referees for their valuable suggestions on the original draft. 相似文献
10.
Duality theory in multiobjective programming 总被引:5,自引:0,他引:5
In this paper, a multiobjective programming problem is considered as that of finding the set of all nondominated solutions with respect to the given domination cone. Two point-to-set maps, the primal map and the dual map, and the vector-valued Lagrangian function are defined, corresponding to the case of a scalar optimization problem. The Lagrange multiplier theorem, the saddle-point theorem, and the duality theorem are derived by using the properties of these maps under adequate convexity assumptions and regularity conditions. 相似文献
11.
A nonsmooth multiobjective optimization problem involving generalized (F, α, ρ, d)-type I function is considered. Karush–Kuhn–Tucker type necessary and sufficient optimality conditions are obtained for a
feasible point to be an efficient or properly efficient solution. Duality results are obtained for mixed type dual under the
aforesaid assumptions. 相似文献
12.
13.
In this article, gap functions for a generalized vector equilibrium problem (GVEP) with explicit constraints are investigated. Under a concept of supremum/infimum of a set, defined in terms of a closure of the set, three kinds of conjugate dual problems are investigated by considering the different perturbations to GVEP. Then, gap functions for GVEP are established by using the weak and strong duality results. As application, the proposed approach is applied to construct gap functions for a vector optimization problem and a generalized vector variational inequality problem. 相似文献
14.
《Optimization》2012,61(3-4):315-324
In this paper, we are concerned with scalarization and the Lagrange duality in multiobjective optimization. After exposing a property of a cone-subconvexlike function, we prove two theorems on scalarization and three theorems of the Lagrange duality. 相似文献
15.
D. T. Luc 《Journal of Optimization Theory and Applications》1987,53(2):303-307
McLinden's result on the equivalence between a theorem of the alternative and a duality theorem for constrained optimization problems is extended to the multiobjective case. We then discuss some existing results on that topic and present an alternative approach to duality relations in conditionally complete lattices.The author is indebted to the referee for his useful remarks on a previous version of the paper. 相似文献
16.
In this paper, we present several conditions for the existence of a Lagrange multiplier or a weak saddle point in multiobjective optimization. Relations between a Lagrange multiplier and a weak saddle point are established. A sufficient condition is also given for the equivalence of the Benson proper efficiency and the Borwein proper efficiency.This research was supported by NSFC under Grant No. 78900011 and by BMADIS. The authors are grateful to two referees for supplying valuable comments and pointing out detailed corrections to the draft paper. The authors also wish to thank Dr. P. L. Yu for valuable comments and suggestions.The revised version of this paper was completed while the second author visited the Faculty of Technical Mathematics and Informatics, Delft University of Technology, Delft, The Netherlands. 相似文献
17.
In this work we characterize objective functions which do not change the set of efficient solutions (weakly efficient solutions, properly efficient solutions). Necessary and sufficient conditions for an objective function to be weakly nonessential (properly nonessential) are presented. We establish relations between weakly nonessential, properly nonessential and nonessential functions. 相似文献
18.
G. J. Zalmai 《Journal of Optimization Theory and Applications》1996,90(2):435-456
We establish necessary and sufficient conditions for properly efficient solutions of a class of nonsmooth nonconvex optimal control problems with multiple fractional objective functions, linear dynamics, and nonlinear inequality constraints on both the state and control variables. Subsequently, we utilize these proper efficiency criteria to construct two multiobjective dual problems and prove appropriate duality theorems. Also, we specialize and discuss these results for a particular case of our principal problem which contains square roots of positivesemidefinite quadratic forms. As special cases of the main proper efficiency and duality results, this paper also contains similar results for control problems with multiple, fractional, and ordinary objective functions. 相似文献
19.
We give a new existence theorem for loose saddle point of set-valued map having values in a partially ordered topological vector space which is based on continuity and quasiconvexity- quasiconcavity of its scalarized maps. Moreover, we prove a new saddle point theorem for vector-valued functions in locally convex topological vector spaces under weak condition that is the semicontinuity of two function scalarization. 相似文献
20.
U. Pursiheimo 《Journal of Optimization Theory and Applications》1980,31(2):261-275
A target moves in Euclideann-spaceR
n according to the generalized conditionally deterministic law. The search density that accumulates on the target during its route determines the probability of detection. A necessary and sufficient condition for the search density (x, t) to be optimal is first represented, when there are two types of constraints for the search density: pointwise constraints and total-amount constraints. The second part consists of formulation of the dual problem with the aid of sensitivity parameters for the constraints. By using the dual functional, we obtain the maximal error from the minimum value of the primal objective functional for an arbitrary feasible . Finally, we study the discretized case, which is necessary for numerical calculations. 相似文献