Some NP-complete problems in quadratic and nonlinear programming
Katta G. Murty,Santosh N. Kabadi +1 more
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A special class of indefinite quadratic programs is constructed, with simple constraints and integer data, and it is shown that checking (a) or (b) on this class is NP-complete.Abstract:
In continuous variable, smooth, nonconvex nonlinear programming, we analyze the complexity of checking whether(a)a given feasible solution is not a local minimum, and(b)the objective function is not bounded below on the set of feasible solutions.
We construct a special class of indefinite quadratic programs, with simple constraints and integer data, and show that checking (a) or (b) on this class is NP-complete. As a corollary, we show that checking whether a given integer square matrix is not copositive, is NP-complete.read more
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Numerical Optimization
Jorge Nocedal,Stephen J. Wright +1 more
TL;DR: Numerical Optimization presents a comprehensive and up-to-date description of the most effective methods in continuous optimization, responding to the growing interest in optimization in engineering, science, and business by focusing on the methods that are best suited to practical problems.
DissertationDOI
Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization
TL;DR: In this paper, the authors introduce a specific class of linear matrix inequalities (LMI) whose optimal solution can be characterized exactly, i.e., the optimal value equals the spectral radius of the operator.
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Semidefinite programming relaxations for semialgebraic problems
TL;DR: It is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility and provide a constructive approach for finding bounded degree solutions to the Positivstellensatz.
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Adjustable robust solutions of uncertain linear programs
TL;DR: The Affinely Adjustable Robust Counterpart (AARC) problem is shown to be, in certain important cases, equivalent to a tractable optimization problem, and in other cases, having a tight approximation which is tractable.
Proceedings Article
Revisiting Frank-Wolfe: Projection-Free Sparse Convex Optimization
TL;DR: A new general framework for convex optimization over matrix factorizations, where every Frank-Wolfe iteration will consist of a low-rank update, is presented, and the broad application areas of this approach are discussed.
References
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Computers and Intractability: A Guide to the Theory of NP-Completeness
TL;DR: The second edition of a quarterly column as discussed by the authors provides a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NP-Completeness,” W. H. Freeman & Co., San Francisco, 1979.
Book
Nonlinear Programming: Sequential Unconstrained Minimization Techniques
TL;DR: This report gives the most comprehensive and detailed treatment to date of some of the most powerful mathematical programming techniques currently known--sequential unconstrained methods for constrained minimization problems in Euclidean n-space--giving many new results not published elsewhere.
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Nonlinear Programming
TL;DR: It is shown that if A is closed for all k → x x, k → y y, where ( k A ∈ ) k y x , then ( ) A ∉ y x .