Application of E. Helly's Theorem to Convex Programming, Problems of Best Approximation and Related Questions

Levin, V. L.

doi:10.1070/sm1969v008n02abeh001118

Cited by 34 publications

(35 citation statements)

References 2 publications

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“…with J(x) := c x) is the possibly non-convex cost J, we show in Appendix A that Helly's dimension ζ for the globally optimal value of SP in (4) can in general be unbounded. Therefore even for the apparently simple non-convex SP in (4) it is impossible to directly apply the classic scenario approach [17], [18], [19], [20] based on Helly's theorem [32], [33].…”

Section: A Known Results On Scenario Approximations Of Chance Constrmentioning

confidence: 99%

A Scenario Approach for Non-Convex Control Design

Grammatico

Zhang

Margellos

et al. 2015

IEEE Trans. Automat. Contr.

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Randomized optimization is an established tool for control design with modulated robustness. While for uncertain convex programs there exist efficient randomized approaches, this is not the case for non-convex problems. Methods based on statistical learning theory are applicable to non-convex problems, but they usually are conservative in achieving the desired probabilistic guarantees. In this paper, we derive a novel scenario approach for a wide class of random non-convex programs, with a sample complexity similar to that of uncertain convex programs and with probabilistic guarantees that hold not only for the optimal solution of the scenario program, but for all feasible solutions inside a set of a-priori chosen complexity. We also address measure-theoretic issues for uncertain convex and non-convex programs. Among the family of non-convex control-design problems that can be addressed via randomization, we apply our scenario approach to stochastic model predictive control for chance constrained nonlinear control-affine systems

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Section: A Known Results On Scenario Approximations Of Chance Constrmentioning

confidence: 99%

A Scenario Approach for Non-Convex Control Design

Grammatico

Zhang

Margellos

et al. 2015

IEEE Trans. Automat. Contr.

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“…We say that the Slater condition holds for K if there exists u ∈ K such that p(u, y) > 0 for all y ∈ S and the point u is called a Slater point. Borwein (1981); Levin (1969)) Assume that the Slater condition holds for K and the index set S is compact in the problem (P). Then for any convex f (x) ∈ R[x], there exist points y 1 , .…”

Section: Notation and Preliminariesmentioning

confidence: 99%

On semi-infinite systems of convex polynomial inequalities and polynomial optimization problems

Guo

Sun

2020

Comput Optim Appl

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We consider the semi-infinite system of polynomial inequalities of the formwhere p(x, y) is a real polynomial in the variables x and the parameters y, the index set S is a basic semialgebraic set in R n , −p(x, y) is convex in x for every y ∈ S. We propose a procedure to construct approximate semidefinite representations of K. There are two indices to index these approximate semidefinite representations. As two indices increase, these semidefinite representation sets expand and contract, respectively, and can approximate K as closely as possible under some assumptions. In some special cases, we can fix one of the two indices or both. Then, we consider the optimization problem of minimizing a convex polynomial over K. We present an SDP relaxation method for this optimization problem by similar strategies used in constructing approximate semidefinite representations of K. Under certain assumptions, some approximate minimizers of the optimization problem can also be obtained from the SDP relaxations. In some special cases, we show that the SDP relaxation for the optimization problem is exact and all minimizers can be extracted.

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“…In particular we can consider measures of the form µ = m i=1 λ i δ(ω i ), i.e., measures with finite support (which is a subset of {ω 1 , ..., ω m } consisting of points ω i such that λ i = 0). For such measure µ we have that In that respect we have the following, quite a nontrivial, result due to Levin [20] (see also [5]). …”

Section: Proof Suppose That Val(p ) = Val(d)mentioning

confidence: 99%

“…If, moreover, (P ) is convex, then there exists such discretization with the following bounds on m. Recall that Helly's theorem says that if A i , i ∈ I, is a finite family of convex subsets of R n such that the intersection of any n + 1 sets of this family is nonempty, then ∩ i∈I A i is nonempty (use of Helly's theorem to derive such bounds for semi-infinite programs seemingly is going back to [20]). Theorem 3.3 Suppose that problem (P ) is convex and reducible.…”

Section: Proof Suppose That Val(p ) = Val(d)mentioning

confidence: 99%