Duality for Stochastic Programming Interpreted as L. P. in $L_p $-Space

Eisner, Mark; Olsen, Paul

doi:10.1137/0128064

Cited by 41 publications

(14 citation statements)

References 14 publications

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“…Such restrictions also appear in the related work of Eisner and Olsen [5,63,Dynkin [4] and Evstigneev [7, 81. (They are partially skirted by Hiriart-Urruty [9] because he deals with the nonconvex case and does not seek any duality relations.) Here we go a long way towards removing these boundedness conditions.…”

Section: (Accepted For Publication September 15 1982)mentioning

confidence: 72%

Deterministic and stochastic optimization problems of bolza type in discrete time

Rockafellart

Wetst

1983

Stochastics

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Section: (Accepted For Publication September 15 1982)mentioning

confidence: 72%

Deterministic and stochastic optimization problems of bolza type in discrete time

Rockafellart

Wetst

1983

Stochastics

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“…The following derivation of D for the linear case is useful in clarifying the relations between the results of this paper and those of [2] and [14]. Let /io(*i) = c, x *,; / 20 (s, *i,…”

Section: Jsmentioning

confidence: 95%

“…Furthermore, for each (x u x 2 ) in R n x R" 2 the functions f 2ι ( , x u x 2 ) are measurable on 5, in fact summable for i = 0 and bounded for / = 1, , m 2 . These assumptions imply that if %,: S -»R n ' and x 2 : S-> R" 2 then F(x, u) is the expected cost (1.3); otherwise F(x, w) = +00. We denote by P(u) the problem of minimizing F(x, u) over all JC E X, i.e.…”

Section: Jsmentioning

confidence: 99%

Stochastic convex programming: basic duality

Rockafellar¹,

Wets²

1976

Pacific J. Math.

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A duality theory is developed for stochastic programs with convex objective and convex constraints. The problem consists in selecting x t E R n * and x 2 E ££°°(S, S, σ R"*) so as to satisfy the constraints and minimize total expected cost, where σ is a probability measure and the constraints as well as the objective are functions of the random elements of the problem. Under the additional restriction that JCI and x 2 (s) belong to compact subsets of R n ' and R "* respectively, it is shown that the problem is equivalent to the more common dynamic formulation for stochastic programs with recourse, a basic duality theorem -of the type min = sup -is proved and qualitative results on the existence of dual solutions are derived.

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“…In this section, we apply a two-stage LDR to the dual of MSLP, with the goal of obtaining lower bounds on the optimal value of MSLP. The dual of MSLP, which we refer to as D-MSLP, is the problem (see [18]):…”

Section: Dual Two-stage Linear Decision Rulesmentioning

confidence: 99%

Two-stage linear decision rules for multi-stage stochastic programming

Bodur

Luedtke

2018

Math. Program.

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Multi-stage stochastic linear programs (MSLPs) are notoriously hard to solve in general. Linear decision rules (LDRs) yield an approximation of an MSLP by restricting the decisions at each stage to be an affine function of the observed uncertain parameters. Finding an optimal LDR is a static optimization problem that provides an upper bound on the optimal value of the MSLP, and, under certain assumptions, can be formulated as an explicit linear program. Similarly, as proposed by Kuhn, Wiesemann, and Georghiou ("Primal and dual linear decision rules in stochastic and robust optimization" Math. Program. 130, 177-209, 2011) a lower bound for an MSLP can be obtained by restricting decisions in the dual of the MSLP to follow an LDR. We propose a new approximation approach for MSLPs, two-stage LDRs. The idea is to require only the state variables in an MSLP to follow an LDR, which is sufficient to obtain an approximation of an MSLP that is a two-stage stochastic linear program (2SLP). We similarly propose to apply LDR only to a subset of the variables in the dual of the MSLP, which yields a 2SLP approximation of the dual that provides a lower bound on the optimal value of the MSLP. Although solving the corresponding 2SLP approximations exactly is intractable in general, we investigate how approximate solution approaches that have been developed for solving 2SLP can be applied to solve these approximation problems, and derive statistical upper and lower bounds on *

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Duality for Stochastic Programming Interpreted as L. P. in $L_p $-Space

Cited by 41 publications

References 14 publications

Deterministic and stochastic optimization problems of bolza type in discrete time

Deterministic and stochastic optimization problems of bolza type in discrete time

Stochastic convex programming: basic duality

Two-stage linear decision rules for multi-stage stochastic programming

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