CONDITIONS FOR GLOBAL OPTIMALITY OF QUADRATIC MINIMIZATION PROBLEMS WITH LMI CONSTRAINTS

2007 ◽  
Vol 24 (02) ◽  
pp. 149-160 ◽  
Author(s):  
VAITHILINGAM JEYAKUMAR ◽  
ZHIYOU WU
Keyword(s):  
Optimality Conditions ◽  
Sufficient Conditions ◽  
Lagrangian Function ◽  
Linear Matrix ◽  
Global Optimality ◽  
Sufficient Optimality Condition ◽  
Box Constraints ◽  
Quadratic Programs ◽  
Quadratic Minimization ◽  
Binary Constraints

In this paper we present sufficient conditions for global optimality of non-convex quadratic programs involving linear matrix inequality (LMI) constraints. Our approach makes use of the concept of a quadratic subgradient. We develop optimality conditions for quadratic programs with LMI constraints by using Lagrangian function and by examining conditions which minimizes a quadratic subgradient of the Lagrangian function over simple bounding constraints. As applications, we obtain sufficient optimality condition for quadratic programs with LMI and box constraints by minimizing a quadrtic subgradient over box constraints. We also give optimality conditions for quadratic minimization involving LMI and binary constraints.

scholarly journals Global Sufficient Optimality Conditions for a Special Cubic Minimization Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
Xiaomei Zhang ◽  
Yanjun Wang ◽  
Weimin Ma
Keyword(s):  
Optimality Conditions ◽  
Minimization Problem ◽  
Sufficient Conditions ◽  
Sufficient Optimality Conditions ◽  
Global Optimality ◽  
Global Optimality Conditions ◽  
Box Constraints ◽  
Global Minimizers ◽  
Quadratic Minimization ◽  
Binary Constraints

We present some sufficient global optimality conditions for a special cubic minimization problem with box constraints or binary constraints by extending the global subdifferential approach proposed by V. Jeyakumar et al. (2006). The present conditions generalize the results developed in the work of V. Jeyakumar et al. where a quadratic minimization problem with box constraints or binary constraints was considered. In addition, a special diagonal matrix is constructed, which is used to provide a convenient method for justifying the proposed sufficient conditions. Then, the reformulation of the sufficient conditions follows. It is worth noting that this reformulation is also applicable to the quadratic minimization problem with box or binary constraints considered in the works of V. Jeyakumar et al. (2006) and Y. Wang et al. (2010). Finally some examples demonstrate that our optimality conditions can effectively be used for identifying global minimizers of the certain nonconvex cubic minimization problem.

scholarly journals Sufficient global optimality conditions for multi-extremal smooth minimisation problems with bounds and linear matrix inequality constraints

2006 ◽  
Vol 47 (4) ◽  
pp. 439-450 ◽  
Author(s):  
N. Q. Huy ◽  
V. Jeyakumar ◽  
G. M. Lee
Keyword(s):  
Linear Matrix Inequality ◽  
Optimality Conditions ◽  
Model Problem ◽  
Inequality Constraints ◽  
Linear Matrix ◽  
Feasible Region ◽  
Global Optimality ◽  
Global Optimality Conditions ◽  
Matrix Inequality ◽  
Analysis And Design

AbstractIn this paper, we present sufficient conditions for global optimality of a general nonconvex smooth minimisation model problem involving linear matrix inequality constraints with bounds on the variables. The linear matrix inequality constraints are also known as “semidefinite” constraints which arise in many applications, especially in control system analysis and design. Due to the presence of nonconvex objective functions such minimisation problems generally have many local minimisers which are not global minimisers. We develop conditions for identifying global minimisers of the model problem by first constructing a (weighted sum of squares) quadratic underestimator for the twice continuously differentiable objective function of the minimisation problem and then by characterising global minimisers of the easily tractable underestimator over the same feasible region of the original problem. We apply the results to obtain global optimality conditions for optinusation problems with discrete constraints.

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