Theory of linear and integer programming
Theory of linear and integer programming
SIAM Review
An exact duality theory for semidefinite programming and its complexity implications
Mathematical Programming: Series A and B
Semidefinite programming in combinatorial optimization
Mathematical Programming: Series A and B - Special issue: papers from ismp97, the 16th international symposium on mathematical programming, Lausanne EPFL
Global Optimization with Polynomials and the Problem of Moments
SIAM Journal on Optimization
On the Complexity of Semidefinite Programs
Journal of Global Optimization
Matrix Sum-of-Squares Relaxations for Robust Semi-Definite Programs
Mathematical Programming: Series A and B
Positive Trigonometric Polynomials and Signal Processing Applications
Positive Trigonometric Polynomials and Signal Processing Applications
Strong duality and minimal representations for cone optimization
Computational Optimization and Applications
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Farkas' lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear matrix inequalities. We provide nonlinear algebraic certificates for all infeasible linear matrix inequalities in the spirit of real algebraic geometry: A linear matrix inequality $Ax\succeq 0$ is infeasible if and only if -1 lies in the quadratic module associated to A. We also present a new exact duality theory for semidefinite programming, motivated by the real radical and sums of squares certificates from real algebraic geometry.