The marginal value formula on regions of stability

Rooyen, M. van; Zlobec, Sanjo

doi:10.1080/02331939308843869

Cited by 3 publications

(6 citation statements)

References 13 publications

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“…(For details over a "region of stability" see [16,Theorem 4.3].) For the sake of simplicity, we will look only at a simple case of the marginal value formula, describe a simple numerical method, and then show how the method can solve Zermelo's problems.…”

Section: (71)mentioning

confidence: 99%

“…We recall (see [25,Theorem 2.2]) that under the assumption that the set of optimal solutions of P(O k) is nonempty and bounded and the mapping F is lower semicontinuous at O k relative to 9 c, there where x kt = x kl (0 ~t) is an optimal solution of the program P(Okt). (For conditions that guarantee the existence of these limits see, e.g., [16]. Also we denote z kl = (x kt, OA:l).…”

Section: Where O K E U and X K = Xk(o K)mentioning

confidence: 99%

“…The PCP version of the marginal value formula (adapted from [16,26]), at the k-th iteration, follows.…”

Section: F~'o--+ F~(o)= {X" Fi(xo)=o Imentioning

confidence: 99%

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Partly convex programming and zermelo's navigation problems

Zlobec

1995

J Glob Optim

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Abstract. Mathematical programs, that become convex programs after "freezing" some variables, are termed partly convex. For such programs we give saddle-point conditions that are both necessary and sufficient that a feasible point be globally optimal. The conditions require "cooperation" of the feasible point tested for optimality, an assumption implied by lower semicontinuity of the feasible set mapping. The characterizations are simplified if certain point-to-set mappings satisfy a "sandwich condition". The tools of parametric optimization and basic point-to-set topology are used in formulating both optimality conditions and numerical methods. In particular, we solve a large class of Zerrnelo's navigation problems and establish global optimality of the numerical solutions. (1991). 90C25, 90C30, 90C90. Mathematics Subject Classifications

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Section: (71)mentioning

confidence: 99%

Section: Where O K E U and X K = Xk(o K)mentioning

confidence: 99%

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Partly convex programming and zermelo's navigation problems

Zlobec

1995

J Glob Optim

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“…For a study of this region in linear programming see [7], for its application in random decision systems see [34]. A characterization of the region of stability is given in [31]. The key to the successful practical solution of an SPP problem is the ability to construct the region of stability (or its subsets) at a given input O .…”

Section: Definition Of Stable Parametric Programmingmentioning

confidence: 99%

“…In Section 4 we find new conditions when a well-known marginal value formula, see, e.g. [31,381, is valid in the absence of a constraint qualification. We show how this formula can be used in solving SPP by linear perturbations.…”

Section: Introductionmentioning

confidence: 98%

Stable parametric programming^*

Zlobec¹

1999

Optimization

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Stable parametric programs (abbreviation: SPP) are parametric programs with a particular continuity (stability) requirement. Optimal solutions of SPP are paths in the space of "parameters" (inputs, data) that preserve continuity of the feasible set point-toset mapping in the space of "decision variables". The end points of these paths optimize the optimal value function on a region of stability.In this paper we study only convex SPP. First we study optimality conditions. If the constraints enjoy the locally-flat-surface ("LFS") property in the decision variable component, then the usual separation arguments apply and we can characterize local and global optimal solutions. Then we consider a well-known marginal value formula for the optimal value function. We prove the formula under new assumptions and then use it to modify a class of quasi-Newton methods in order to solve convex SPP. Finally, several solved case are reported.

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