A Variable-Complexity Norm Maximization Problem

Mangasarian, O. L.; Shiau, Tzong-Huei

doi:10.1137/0607052

Cited by 31 publications

(8 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In that case, Z N is polyhedral. The decision problem associated to the maximization of the L2-norm of a vector over a polyhedral set is known to be NP-complete [28]. Proposition 2 indicates that we should not expect to develop exact solution algorithms for our problem.…”

Section: Proposition 2 (Np-completeness) the Decision Problem Associmentioning

confidence: 99%

Optimal Information Blending with Measurements in the L² Sphere

2015

View full text Add to dashboard Cite

We consider a sequential information collection problem where a risk-averse decision-maker updates a Bayesian belief about the unknown objective function of a linear program. The information is collected in the form of a linear combination of the objective coefficients, subject to random noise. We have the ability to choose the weights in the linear combination, creating a new, nonconvex continuous optimization problem, which we refer to as information blending. We develop two optimal blending strategies: an active learning method that maximizes uncertainty reduction, and an economic approach that maximizes an expected improvement criterion. Semidefinite programming relaxations are used to create efficient convex approximations to the nonconvex blending problem.

show abstract

Section: Proposition 2 (Np-completeness) the Decision Problem Associmentioning

confidence: 99%

Optimal Information Blending with Measurements in the L² Sphere

2015

View full text Add to dashboard Cite

show abstract

“…Hence, the question how efficiently these functionals can be computed or approximated has been studied extensively, e.g. in [1,2,13,15,19]. Of particular interest is the problem of maximizing (the p-th power of) a p-norm over a polytope.…”

Section: Introductionmentioning

confidence: 99%

“…

The problem of maximizing the p-th power of a p-norm over a halfspace-presented polytope in R d is a convex maximization problem which plays a fundamental role in computational convexity. It has been shown in [19] that this problem is NP-hard for all values p ∈ N, if the dimension d of the ambient space is part of the input. In this paper, we use the theory of parametrized complexity to analyze how heavily the hardness of norm maximization relies on the parameter d. More precisely, we show that for p = 1 the problem is fixed parameter tractable but that for all p ∈ N \ {1} norm maximization is W[1]-hard.

…”

mentioning

confidence: 99%

Fixed-Parameter Complexity and Approximability of Norm Maximization

Knauer

König

Werner

2015

Discrete Comput Geom

View full text Add to dashboard Cite

Abstract.The problem of maximizing the p-th power of a p-norm over a halfspace-presented polytope in R d is a convex maximization problem which plays a fundamental role in computational convexity. It has been shown in [19] that this problem is NP-hard for all values p ∈ N, if the dimension d of the ambient space is part of the input. In this paper, we use the theory of parametrized complexity to analyze how heavily the hardness of norm maximization relies on the parameter d. More precisely, we show that for p = 1 the problem is fixed parameter tractable but that for all p ∈ N \ {1} norm maximization is W[1]-hard. Concerning approximation algorithms for norm maximization, we show that for fixed accuracy, there is a straightforward approximation algorithm for norm maximization in FPT running time, but there is no FPT approximation algorithm, the running time of which depends polynomially on the accuracy. As with the NP-hardness of norm maximization, the W[1]-hardness immediately carries over to various radius computation tasks in Computational Convexity.

show abstract

“…The 2-norm maximization problem and its related decision problem ( 1 ) are shown to be NP-complete in Mangasarian and Shiau [20]. Define the parameters of ( ) as…”

Section: Theorem 22 For An Arbitrary X ∈ X There Exists a Sequencementioning

confidence: 99%

“…For the minimax problem, these probabilities were obtained from the optimal dual variables to the semidefinite optimization problem (20). For the data-driven approach, the probabilities were obtained through an extensive simulation using 100 000 samples from the normal distribution.…”

Section: Numerical Examplementioning

confidence: 99%

Models for Minimax Stochastic Linear Optimization Problems with Risk Aversion

et al. 2010

View full text Add to dashboard Cite

We propose a semidefinite optimization (SDP) model for the class of minimax two-stage stochastic linear optimization problems with risk aversion. The distribution of second-stage random variables belongs to a set of multivariate distributions with known first and second moments. For the minimax stochastic problem with random objective, we provide a tight SDP formulation. The problem with random right-hand side is NP-hard in general. In a special case, the problem can be solved in polynomial time. Explicit constructions of the worst-case distributions are provided. Applications in a productiontransportation problem and a single facility minimax distance problem are provided to demonstrate our approach. In our experiments, the performance of minimax solutions is close to that of data-driven solutions under the multivariate normal distribution and better under extremal distributions. The minimax solutions thus guarantee to hedge against these worst possible distributions and provide a natural distribution to stress test stochastic optimization problems under distributional ambiguity.

show abstract

A Variable-Complexity Norm Maximization Problem

Cited by 31 publications

References 10 publications

Optimal Information Blending with Measurements in the L² Sphere

Optimal Information Blending with Measurements in the L² Sphere

Fixed-Parameter Complexity and Approximability of Norm Maximization

Models for Minimax Stochastic Linear Optimization Problems with Risk Aversion

Contact Info

Product

Resources

About

A Variable-Complexity Norm Maximization Problem

Cited by 31 publications

References 10 publications

Optimal Information Blending with Measurements in the L2 Sphere

Optimal Information Blending with Measurements in the L2 Sphere

Fixed-Parameter Complexity and Approximability of Norm Maximization

Models for Minimax Stochastic Linear Optimization Problems with Risk Aversion

Contact Info

Product

Resources

About

Optimal Information Blending with Measurements in the L² Sphere

Optimal Information Blending with Measurements in the L² Sphere