R. Henrion and W. Romisch, Metric regularity and quantitative stability in stochastic programs with probabilistic constraints, Mathematical Programming 84 (1999), 55-88. Home/Search Document Details and Download
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Quantitative stability in stochastic programming: The method.. - Rachev, Römisch (2000) (2 citations) (Correct)
.... programming (see e.g. 13] 22] 54] 3] 36] and [30] for qualitative results, and [20] 21] 19] 49] 15] and [50] for rates of convergence, large deviation results and central limit theorems) In the present paper, we take up the idea of using bounds for empirical processes (raised in [16] and [29] and extend some of the earlier work. Our paper is organized as follows. Section 2 contains the general perturbation results for (1) together with a discussion of how to associate canonical metrics with more speci c stochastic programs. In Section 3 we discuss linear two stage, ....
....j ( x n ) n (d ) 0 : and, hence, x 2 M U ( To obtain quantitative stability results for (1) a stability property of the constraint set M( when perturbing the probabilistic constraints is needed. Consistently with the general de nition of metric regularity for multifunctions (see e.g. [38, 16]) we consider the set valued mapping y 7 M y ( from IR d to IR m , where M y ( fx 2 X : Z f j ( x) d ) y j ; j = 1; dg; and say that its inverse x 7 M 1 x ( fy 2 IR d : x 2 M y ( g is metrically regular at some pair ( x; 0) 2 IR m IR d with x 2 M( ....
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R. Henrion and W. Romisch, Metric regularity and quantitative stability in stochastic programs with probabilistic constraints, Mathematical Programming 84 (1999), 55-88.
Stability of Solutions to Chance Constrained Stochastic Programs - Henrion, Römisch (Correct)
....P(IR s ) of all Borel probability measures on IR s . Here we equip this space with the uniform or Kolmogorov distance dK ( kF Gamma F k1 = sup y2IR s jF (y) Gamma F (y)j: Stability issues for chance constrained programs are adressed in a number of papers (see e.g. 1] 4] 5] [6], 13] 15] and references therein) A typical question in this respect is the continuity behaviour of optimal values ( inffg(x) j x 2 X; F (h(x) pg and solution sets Psi( argminfg(x) j x 2 X; F (h(x) pg of problem (3) when the measure is subjected to variations in (P(IR s ) ....
....(3) when the measure is subjected to variations in (P(IR s ) dK ) In the present paper, we look at conditions on model (3) implying quantitative continuity properties of solution sets with respect to the metric dK . Our main result (Theorem 2.5) extends our earlier work (Theorem 4. 3 in [6]) for the linear quadratic case (i.e. g convex quadratic, h linear and X convex polyhedral) considerably. It provides conditions implying upper Holder continuity of solution sets at the original measure with some rate that depends essentially on the data g; h and X. Our stability results are ....
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R. Henrion and W. Romisch, "Metric regularity and quantitative stability in stochastic programs with probabilistic constraints," Humboldt-Universitat Berlin, Institut fur Mathematik, Preprint Nr. 96-2, 1996, and submitted to Mathematical Programming.
Regular Castaing Representations of Multifunctions with.. - Dentcheva (1998) (Correct)
....Furthermore, let kfk L be the usual Lipschitz norm: kfk L = kfk1 sup x;y2Y jf(x) Gamma f(y)j kx Gamma yk : It is known (cf. 14] that (M p (X) W p ) is a metric space. Quantitative stability of stochastic programs with respect to perturbations of probability measures is investigated in [15, 16, 26, 27, 28, 29]. We shall utilize some of the results presented in those papers. 5.1 Stochastic Recourse Programs Let us consider a two stage stochastic program with linear recourse and random right hand side: minfg(x) Q (Ax) x 2 Cg (7) Q ( Z IR Q( Gamma ) d ) 8) Q(z) minfq y : ....
....U of ( and ffi 0 such that setting : U IR n as ( U ( where U = f 2 P(IR s ) ff( ffig, it holds that the mapping admits a Castaing representation by Steiner selections which are Holder continuous of order 1=2 at . Proof: We apply Theorem 4. 3 in [16]. Under the assumption of the proposition, there are constants L 0 such that: dH ( U ( Lff( 1=2 (14) for any probability measure 2 U . Using the notation for the restriction of the solution set mapping to the mapping of local minimizers the above inequality means that ....
R.Henrion and W.Romisch, Metric regularity and quantitative stability in stochastic programs with probabilistic constraints, Mathematical Programming, to appear.
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