Accuracy Certificates for Computational Problems with Convex Structure

Nemirovski, Arkadi; Onn, Shmuel; Rothblum, Uriel G.

doi:10.1287/moor.1090.0427

Cited by 44 publications

(59 citation statements)

References 16 publications

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“…The convergence rate is exponential, but there is a direct and strong dependence on the dimension of the problem p. Note that dual certificates may be obtained at limited additional computational cost, with no extra information [168]. This algorithm is typically slow in practice since it has a running-time of O(p 3 ) per iteration.…”

Section: This Implies That There Exists I T Such That W I ∈ K Andmentioning

confidence: 99%

Learning with Submodular Functions: A Convex Optimization Perspective

Bach

2013

FNT in Machine Learning

211

226

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Section: This Implies That There Exists I T Such That W I ∈ K Andmentioning

confidence: 99%

Learning with Submodular Functions: A Convex Optimization Perspective

Bach

2013

FNT in Machine Learning

211

226

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“…Assume that (17), (18) and (20) take place. Renaming the variables according to (14), (15) is the dual SP problem associated with the bilinear master SP problem…”

Section: Discussionmentioning

confidence: 99%

“…2. We solve (P) to a desired accuracy by first-order algorithm producing accuracy certificates [20] and use these certificates to recover an approximate solution of required accuracy to the problem of interest.…”

Section: Induced Convex-concave Functionsmentioning

confidence: 99%

“…calls to the oracle, the algorithm produces a t-step execution protocol associated with θ , U , V . The importance of these notions in our context stems from the following simple observation [20]:…”

Section: Preliminaries: Execution Protocols Accuracy Certificates Rmentioning

confidence: 99%

“…, w L ), and L =1 L (η) is the linear function g, η , g = g , and thus is convex. It is known (see, for example, [20]) that Nash equilibria in convex Nash problem are exactly the weak solutions to the VI given by monotone operator…”

Section: Nash Equilibrium With Pairwise Interactionsmentioning

confidence: 99%

See 2 more Smart Citations

Decomposition Techniques for Bilinear Saddle Point Problems and Variational Inequalities with Affine Monotone Operators

Cox

Juditsky

Nemirovski

2016

J Optim Theory Appl

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International audienceThe majority of first-order methods for large-scale convex–concave saddle point problems and variational inequalities with monotone operators are proximal algorithms. To make such an algorithm practical, the problem’s domain should be proximal-friendly—admit a strongly convex function with easy to minimize linear perturbations. As a by-product, this domain admits a computationally cheap linear minimization oracle (LMO) capable to minimize linear forms. There are, however, important situations where a cheap LMO indeed is available, but the problem domain is not proximal-friendly, which motivates search for algorithms based solely on LMO. For smooth convex minimization, there exists a classical algorithm using LMO—conditional gradient. In contrast, known to us similar techniques for other problems with convex structure (nonsmooth convex minimization, convex–concave saddle point problems, even as simple as bilinear ones, and variational inequalities with monotone operators, even as simple as affine) are quite recent and utilize common approach based on Fenchel-type representations of the associated objectives/vector fields. The goal of this paper was to develop alternative (and seemingly much simpler) decomposition techniques based on LMO for bilinear saddle point problems and for variational inequalities with affine monotone operators

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Non‐zero‐sum nonlinear network path interdiction with an application to inspection in terror networks

Goldberg

2017

Naval Research Logistics

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A simultaneous non‐zero‐sum game is modeled to extend the classical network interdiction problem. In this model, an interdictor (e.g., an enforcement agent) decides how much of an inspection resource to spend along each arc in the network to capture a smuggler. The smuggler (randomly) selects a commodity to smuggle—a source and destination pair of nodes, and also a corresponding path for traveling between the given pair of nodes. This model is motivated by a terrorist organization that can mobilize its human, financial, or weapon resources to carry out an attack at one of several potential target destinations. The probability of evading each of the network arcs nonlinearly decreases in the amount of resource that the interdictor spends on its inspection. We show that under reasonable assumptions with respect to the evasion probability functions, (approximate) Nash equilibria of this game can be determined in polynomial time; depending on whether the evasion functions are exponential or general logarithmically‐convex functions, exact Nash equilibria or approximate Nash equilibria, respectively, are computed. © 2017 Wiley Periodicals, Inc. Naval Research Logistics 64: 139–153, 2017

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Accuracy Certificates for Computational Problems with Convex Structure

Cited by 44 publications

References 16 publications

Learning with Submodular Functions: A Convex Optimization Perspective

Learning with Submodular Functions: A Convex Optimization Perspective

Decomposition Techniques for Bilinear Saddle Point Problems and Variational Inequalities with Affine Monotone Operators

Non‐zero‐sum nonlinear network path interdiction with an application to inspection in terror networks

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