Necessary and sufficient conditions for S-lemma and nonconvex quadratic optimization

Jeyakumar, V.; Huy, Nguyễn Quang; Li, Guoyin

doi:10.1007/s11081-008-9076-9

Cited by 28 publications

(21 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This provides a convenient way for finding the maximum eigenvalue of an essentially non-negative tensor. For other optimization problems which can be converted to solving a linear semi-definite programming problem see [30,31,32].…”

Section: Clearlymentioning

confidence: 99%

See 1 more Smart Citation

Finding the Maximum Eigenvalue of Essentially Nonnegative Symmetric Tensors via Sum of Squares Programming

et al. 2013

J Optim Theory Appl

Self Cite

View full text Add to dashboard Cite

Finding the maximum eigenvalue of a tensor is an important topic in tensor computation and multilinear algebra. Recently, when the tensor is non-negative in the sense that all of its entries are non-negative, efficient numerical schemes have been proposed to calculate the maximum eigenvalue based on a Perron-Frobenius type theorem for non-negative tensors. In this paper, we consider a new class of tensors called essentially non-negative tensors, which extends the non-negative tensors, and examine the maximum eigenvalue of an essentially non-negative tensor using the polynomial optimization techniques. We first establish that finding the maximum eigenvalue of an essentially non-negative symmetric tensor is equivalent to solving a sum of squares of polynomials (SOS) optimization problem, which, in turn, can be equivalently rewritten as a semi-definite programming problem. Then, using this sum of squares programming problem, we also provide upper as well as lower estimate for the maximum eigenvalue of general symmetric tensors. These upper and lower estimates can be calculated in terms of the entries of the tensor.

show abstract

Section: Clearlymentioning

confidence: 99%

“…Recall that an n × n matrix is called an Z-matrix (see [30,31]) if all its off-diagonal elements are non-positive. Extending this, we say an mth-order n-dimensional tensor A is a Z-tensor if A i 1 ,...,im ≤ 0 for all {i 1 , .…”

Section: Remark 33 (Other Approaches For Finding the Maximum Eigenvmentioning

confidence: 99%

Finding the Maximum Eigenvalue of Essentially Nonnegative Symmetric Tensors via Sum of Squares Programming

et al. 2013

J Optim Theory Appl

Self Cite

View full text Add to dashboard Cite

show abstract

“…Theorems of the alternative for arbitrary finite systems of linear or convex inequalities [1][2][3][4]21,23] have played key roles often in the development of optimaity principles for continuous optimization problems and in the convergence analysis of optimization algorithms. Unlike the theorems of the alternative for linear systems such as the Farkas lemma [1,22] which provides a numerically checkable alternative certificate of the solvability of the given linear system, alternative theorems for convex inequality systems do not, in general, provide such certificates even under a constraint qualification.…”

Section: Introductionmentioning

confidence: 99%

A new class of alternative theorems for SOS-convex inequalities and robust optimization

Jeyakumar

2013

Applicable Analysis

View full text Add to dashboard Cite

In this paper, we present a new class of alternative theorems for SOS-convex inequality systems without any qualifications. This class of theorems provides an alternative equations in terms of sums of squares to the solvability of the given inequality system. A strong separation theorem for convex sets, described by convex polynomial inequalities, plays a key role in establishing the class of alternative theorems. Consequently, we show that the optimal values of various classes of robust convex optimization problems are equal to the optimal values of related semidefinite programming problems (SDPs) and so, the value of the robust problem can be found by solving a single SDP. The class of problems includes programs with SOS-convex polynomials under data uncertainty in the objective function such as uncertain quadratically constrained quadratic programs. The SOS-convexity is a computationally tractable relaxation of convexity for a real polynomial. We also provide an application of our theorem of the alternative to a multi-objective convex optimization under data uncertainty.

show abstract

“…More precisely, when X = R n and P = [0, +∞[ with g being a quadratic function that is not identically zero, the authors in [15] prove that, (P) has strong duality for each quadratic function f if, and only if there existsx ∈ R n such that g(x) < 0.…”

Section: Introduction and Formulation Of The Problemmentioning

confidence: 99%

A complete characterization of strong duality in nonconvex optimization with a single constraint

Flores-Bazán

Vera

2011

J Glob Optim

View full text Add to dashboard Cite

We first establish sufficient conditions ensuring strong duality for cone constrained nonconvex optimization problems under a generalized Slater-type condition. Such conditions allow us to cover situations where recent results cannot be applied. Afterwards, we provide a new complete characterization of strong duality for a problem with a single constraint: showing, in particular, that strong duality still holds without the standard Slater condition. This yields Lagrange multipliers characterizations of global optimality in case of (not necessarily convex) quadratic homogeneous functions after applying a generalized joint-range convexity result. Furthermore, a result which reduces a constrained minimization problem into one with a single constraint under generalized convexity assumptions, is also presented.

show abstract

Necessary and sufficient conditions for S-lemma and nonconvex quadratic optimization

Cited by 28 publications

References 18 publications

Finding the Maximum Eigenvalue of Essentially Nonnegative Symmetric Tensors via Sum of Squares Programming

Finding the Maximum Eigenvalue of Essentially Nonnegative Symmetric Tensors via Sum of Squares Programming

A new class of alternative theorems for SOS-convex inequalities and robust optimization

A complete characterization of strong duality in nonconvex optimization with a single constraint

Contact Info

Product

Resources

About