An efficient algorithm for globally minimizing a quadratic function under convex quadratic constraints

2000 ◽  
Vol 87 (3) ◽  
pp. 401-426 ◽  
Author(s):  
Le Thi Hoai An
Keyword(s):  
Efficient Algorithm ◽  
Quadratic Function ◽  
Quadratic Constraints ◽  
Convex Quadratic Constraints

scholarly journals Off-line robustification of Generalized Predictive Control for uncertain systems

2014 ◽  
Vol 24 (4) ◽  
pp. 499-513 ◽  
Author(s):  
Khelifa Khelifi Otmane ◽  
Nordine Bali ◽  
Lazhari Nezli
Keyword(s):  
Predictive Control ◽  
Uncertain Systems ◽  
Optimization Problem ◽  
Main Idea ◽  
Optimal Solution ◽  
Generalized Predictive Control ◽  
Quadratic Constraints ◽  
Stability Robustness ◽  
Convex Quadratic Constraints ◽  
Polytopic Uncertainties

Abstract An off-line methodology was proposed for enhancing the robustness of an initial Generalized Predictive Control (GPC) by convex optimization of the Youla parameter. However, this procedure of robustification is restricted with the case of the systems affected only by unstructured uncertainties. This paper proposes an extension of this method to the systems subjected to both unstructured and structured polytopic uncertainties. The main idea consists in adding supplementary constraints to the optimization problem which validates the Lipatov stability condition at each vertex of the polytope. These polytopic uncertainties impose a set of non convex quadratic constraints. The globally optimal solution is found by means of the GloptiPoly3 software. Therefore, this robustification provides stability robustness towards unstructured uncertainties for the nominal system, while guaranteeing stability properties over a specified polytopic domain of uncertainties. Finally, an illustrative example is given

scholarly journals The Generalized Trust-Region Sub-Problem with Additional Linear Inequality Constraints—Two Convex Quadratic Relaxations and Strong Duality

Symmetry ◽  
2020 ◽  
Vol 12 (8) ◽  
pp. 1369
Author(s):  
Temadher A. Almaadeed ◽  
Akram Taati ◽  
Maziar Salahi ◽  
Abdelouahed Hamdi
Keyword(s):  
Linear Inequality ◽  
Sufficient Conditions ◽  
Optimal Solution ◽  
Quadratic Function ◽  
Inequality Constraints ◽  
Fixed Number ◽  
Semidefinite Optimization ◽  
Global Optimality ◽  
Quadratic Constraints ◽  
Linear Inequality Constraints

In this paper, we study the problem of minimizing a general quadratic function subject to a quadratic inequality constraint with a fixed number of additional linear inequality constraints. Under a regularity condition, we first introduce two convex quadratic relaxations (CQRs), under two different conditions, that are minimizing a linear objective function over two convex quadratic constraints with additional linear inequality constraints. Then, we discuss cases where the CQRs return the optimal solution of the problem, revealing new conditions under which the underlying problem admits strong Lagrangian duality and enjoys exact semidefinite optimization relaxation. Finally, under the given sufficient conditions, we present necessary and sufficient conditions for global optimality of the problem and obtain a form of S-lemma for a system of two quadratic and a fixed number of linear inequalities.

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