Geometric conditions for Kuhn-Tucker sufficiency of global optimality in mathematical programming

We present geometric criteria for a feasible point that satisfies the Kuhn-Tucker conditions to be a global minimizer of mathematical programming problems with or without bounds on the variables. The criteria apply to multi-extremal programming problems which may have several local minimizers that are not global. We establish such criteria in terms of underestimators of the Lagrangian of the problem. The underestimators are required to satisfy certain geometric property such as the convexity (or a generalized convexity) property. We show that the biconjugate of the Lagrangian can be chosen as a convex underestimator whenever the biconjugate coincides with the Lagrangian at a point. We also show how suitable underestimators can be constructed for the Lagrangian in the case where the problem has bounds on the variables. Examples are given to illustrate our results.

[1]  Panos M. Pardalos,et al.  Quadratic Programming with Box Constraints , 1997 .

[2]  Zhi-You Wu,et al.  Sufficient Global Optimality Conditions for Non-convex Quadratic Minimization Problems With Box Constraints , 2006, J. Glob. Optim..

[3]  Olvi L. Mangasarian,et al.  Nonlinear Programming , 1969 .

[4]  N. Q. Huy,et al.  Kuhn-Tucker sufficiency for global minimum of multi-extremal mathematical programming problems , 2007 .

[5]  Christodoulos A. Floudas,et al.  Deterministic global optimization - theory, methods and applications , 2010, Nonconvex optimization and its applications.

[6]  J. Ben Rosen,et al.  Approximating Data in $$\mathfrak{R}^{n}$$ by a Quadratic Underestimator with Specified Hessian Minimum and Maximum Eigenvalues , 2006, J. Glob. Optim..

[7]  Arnold Neumaier,et al.  An optimality criterion for global quadratic optimization , 1992, J. Glob. Optim..

[8]  P. Pardalos,et al.  Handbook of global optimization , 1995 .

[9]  Morgan A. Hanson.  A Generalization of the Kuhn-Tucker Sufficiency Conditions , 1994 .

[10]  Jean-Baptiste Hiriart-Urruty.  When is a point x satisfying $\nabla f(x) = 0 a global minimum of f ? , 1986 .

[12]  B. Craven.  Control and optimization , 2019, Mathematical Modelling of the Human Cardiovascular System.

[13]  Mokhtar S. Bazaraa,et al.  Nonlinear Programming: Theory and Algorithms , 1993 .

[14]  N. Q. Huy,et al.  Sufficient global optimality conditions for multi-extremal smooth minimisation problems with bounds and linear matrix inequality constraints , 2006, The ANZIAM Journal.

[15]  P. Pardalos,et al.  From local to global optimization , 2001 .

[16]  P. Pardalos,et al.  Optimization in computational chemistry and molecular biology : local and global approaches , 2000 .

[17]  J. Zowe,et al.  Regularity and stability for the mathematical programming problem in Banach spaces , 1979 .

[18]  Arnold Neumaier.  Second-order sufficient optimality conditions for local and global nonlinear programming , 1996, J. Glob. Optim..

[19]  Vaithilingam Jeyakumar,et al.  On generalised convex mathematical programming , 1992, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[20]  C. Adjiman,et al.  A global optimization method, αBB, for general twice-differentiable constrained NLPs—II. Implementation and computational results , 1998 .

[21]  M. A. Hanson.  On sufficiency of the Kuhn-Tucker conditions , 1981 .

[22]  Christodoulos A. Floudas,et al.  Computational Experience with a New Class of Convex Underestimators: Box-constrained NLP Problems , 2004, J. Glob. Optim..

[23]  Zhi-You Wu,et al.  Non-convex quadratic minimization problems with quadratic constraints: global optimality conditions , 2007, Math. Program..