Some NP-complete problems in quadratic and nonlinear programming
Mathematical Programming: Series A and B
Nonlinear optimization: complexity issues
Nonlinear optimization: complexity issues
A semidefinite framework for trust region subproblems with applications to large scale minimization
Mathematical Programming: Series A and B
Interior point algorithms: theory and analysis
Interior point algorithms: theory and analysis
Robust Solutions to Least-Squares Problems with Uncertain Data
SIAM Journal on Matrix Analysis and Applications
On the Rank of Extreme Matrices in Semidefinite Programs and the Multiplicity of Optimal Eigenvalues
Mathematics of Operations Research
Mathematics of Operations Research
Global Optimization with Polynomials and the Problem of Moments
SIAM Journal on Optimization
Robust One-Period Option Hedging
Operations Research
Optimal linear fusion for distributed detection via semidefinite programming
IEEE Transactions on Signal Processing
Semidefinite relaxation bounds for bi-quadratic optimization problems with quadratic constraints
Journal of Global Optimization
Some Equivalent Results with Yakubovich's $S$-Lemma
SIAM Journal on Control and Optimization
Biquadratic Optimization Over Unit Spheres and Semidefinite Programming Relaxations
SIAM Journal on Optimization
SIAM Journal on Optimization
Alternating direction method for bi-quadratic programming
Journal of Global Optimization
Eigenvalue techniques for convex objective, nonconvex optimization problems
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
Journal of Global Optimization
Exploiting equalities in polynomial programming
Operations Research Letters
Maximum Block Improvement and Polynomial Optimization
SIAM Journal on Optimization
SIAM Journal on Optimization
Duality and solutions for quadratic programming over single non-homogeneous quadratic constraint
Journal of Global Optimization
Computational Geometry: Theory and Applications
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We derive linear matrix inequality (LMI) characterizations and dual decomposition algorithms for certain matrix cones which are generated by a given set using generalized co-positivity. These matrix cones are in fact cones of nonconvex quadratic functions that are nonnegative on a certain domain. As a domain, we consider for instance the intersection of a (upper) level-set of a quadratic function and a half-plane. Consequently, we arrive at a generalization of Yakubovich's S-procedure result. Although the primary concern of this paper is to characterize the matrix cones by LMIs, we show, as an application of our results, that optimizing a general quadratic function over the intersection of an ellipsoid and a half-plane can be formulated as semidefinite programming (SDP), thus proving the polynomiality of this class of optimization problems, which arise, e.g., from the application of the trust region method for nonlinear programming. Other applications are in control theory and robust optimization.