Abstract
In this article, by using the Lagrangian function, we investigate the sufficient global optimality conditions for a class of semi-definite optimization problems, where the objective function are general nonlinear, the variables are mixed integers subject to linear matrix inequalities (LMIs) constraints as well as bounded constraints. In addition, the sufficient global optimality conditions for general nonlinear programming problems are derived, where the variables satisfy LMIs constraints and box constraints or bivalent constraints. Furthermore, we give the sufficient global optimality conditions for standard semi-definite programming problem, where the objective function is linear, the variables satisfy linear inequalities constraints and box constraints. Mathematics Subject Classification 2010: 90C30; 90C26; 90C11.
Highlights
As we know semi-definite programming (SDP) can be viewed as a natural extension of linear programming where the componentwise inequalities between vectors are replaced by matrix inequalities
Jeyakumar et al [11] have established some necessary and sufficient conditions for a given feasible point to be a global minimizer of some minimization problems with mixed variables
Jeyakumar and Wu [13] have presented sufficient conditions for global optimality of non-convex quadratic programs involving linear matrix inequality (LMI) constraints by using Lagrangian function and by examining conditions which minimizes a quadratic subgradient of the Lagrangian function over simple bounding constraints
Summary
As we know semi-definite programming (SDP) can be viewed as a natural extension of linear programming where the componentwise inequalities between vectors are replaced by matrix inequalities. Jeyakumar et al [9] have given Lagrange multiplier conditions for global optimality of general quadratic minimization problems with quadratic constraints. Jeyakumar et al [10] have obtained sufficient global optimality conditions for a quadratic minimization problem subject to box constraints or binary constraints. Jeyakumar et al [11] have established some necessary and sufficient conditions for a given feasible point to be a global minimizer of some minimization problems with mixed variables.
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