ON DUALITY THEORY OF CONIC LINEAR PROBLEMS

By Alexander Shapiro. 2001

398 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Management Science and Operations Research, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/0d39f720947211a0a3759e29a5f7302a0b496610

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Citations in the corpus

Distributionally robust joint chance constraints with second-order moment information (2011, neutral): “, [25]).”
Robust sample average approximation (2014, methodology): “Introducing the variables Hk and this equivalent constraint into the problem (58) and invoking [46] we find that (58) is equal to the dual problem:”
Data-driven chance constrained stochastic program (2015, neutral): “Isii [16], Smith [36], and Shapiro [34]).”
Data-Driven DG Capacity Assessment Method for Active Distribution Networks (2017, background): “According to duality theory [26], equation (24) combined with the probability distribution set (12) can be converted into equation (25) using strong duality, whose proof is provided in the Appendix of this paper.”
Data-driven robust optimization (2013, background): “[45] and Shapiro [15] this optimization admits the following strong dual:”
Distributionally Robust Optimization: A Review (2019, methodology): “(Shapiro [289, Definition 2.1]) A (real-valued) risk measure ρ : Z 7→ R is called law invariant if for all Z,Z ′ ∈ Z, Z d∼ Z ′ implies that ρ(Z) = ρ(Z ′).”
Regularization via Mass Transportation (2017, background): “…uniform distribution on the training samples, which implies that Π is completely determined by the conditional distributions The resulting generalized moment problem over the normalized measures Q i admits the semi-inﬁnite dual sup Strong duality holds for any ρ > 0 due to [34, Proposition 3.4].”
Optimal Inequalities in Probability Theory: A Convex Optimization Approach (2005, background): “Table 1 summarizes the contributions of the paper to the ( n, k, Ω)-bound problem.”
Distributionally Robust Convex Optimization (2014, background): “Contrary to robust optimization, stochastic programming explicitly accounts for distributional information through expectation constraints of the form EQ0 [v(x, z̃)] ≤ w, (2) where z̃ ∈ RP is now a random vector, and the expectation is taken with respect to the distribution Q0 of z̃.”
Robust SAA (2014, background): “4 of [35] requires a generalized Slater point.”
Distributionally Robust Logistic Regression (2015, neutral): “=1 si s.t. sup x∈Rn lβ(x,+1) −λkxˆi −xk − 1 2λκ(1−yˆi) ≤ si ∀i ≤ N sup x∈Rn lβ(x,−1) −λkxˆi −xk − 1 2λκ(1+ ˆyi) ≤ si ∀i ≤ N λ ≥ 0 (11d) Strong duality holds for any ε > 0 due to Proposition 3.4 in [34]. Lemma 1 then implies min β,λ,si λε+ 1 N PN i=1 si s.t. lβ(ˆxi,+1)− 1 2λκ(1−yˆi) ≤ si ∀i”
Distributionally Robust Co-Optimization of Energy and Reserve Dispatch (2016, background): “1) can be derived in the spirit of the duality theory of conic linear programming [34], [36] and shown below”
Sparsity of solutions for variational inverse problems with finite-dimensional data (2018, background): “First of all we recover the well-known result (see for example [30]) that minimizing the Radon norm of a measure under a finite-dimensional data constraints one recovers a minimizer that is made of delta peaks.”
Robust Wasserstein profile inference and applications to machine learning (2016, background): “The approximate representation for regularized logistic regression in [42] is based on the semi-infinite linear programming duality results due to [44].”