Global optimality conditions for cubic minimization problems with cubic constraints

In this paper, we present global optimality conditions for cubic minimization involving cubic constraints and box or bivalent constraints, where the cubic objective function and cubic constraints contain no cross terms. By utilizing quadratic underestimators, we first derive sufficient global optimality conditions for a global minimizer of cubic minimization problems with cubic inequality and box constraints. Then we establish them for cubic minimization with cubic inequality and bivalent constraints. Finally, we establish sufficient and necessary global optimality condition for cubic minimization with cubic equality and binary constraints.

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

[2]  Vaithilingam Jeyakumar,et al.  Lagrange multiplier necessary conditions for global optimality for non-convex minimization over a quadratic constraint via S-lemma , 2009, Optim. Lett..

[3]  Guoyin Li,et al.  Global Quadratic Minimization over Bivalent Constraints: Necessary and Sufficient Global Optimality Condition , 2012, J. Optim. Theory Appl..

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

[5]  R. Horst,et al.  Necessary and Sufficient Global Optimality Conditions for Convex Maximization Revisited , 1998 .

[6]  Geir Dahl,et al.  A note on diagonally dominant matrices , 2000 .

[7]  Roummel F. Marcia,et al.  Iterative Convex Quadratic Approximation for Global Optimization in Protein Docking , 2005, Comput. Optim. Appl..

[8]  P. Parrilo.  Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization , 2000 .

[9]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

[10]  Liansheng Zhang,et al.  Global optimality conditions for quadratic 0-1 optimization problems , 2010, J. Glob. Optim..

[11]  R. Canfield.  Multipoint cubic surrogate function for sequential approximate optimization , 2004 .

[12]  Vaithilingam Jeyakumar,et al.  Global optimality principles for polynomial optimization over box or bivalent constraints by separable polynomial approximations , 2014, J. Glob. Optim..

[13]  Yurii Nesterov,et al.  Accelerating the cubic regularization of Newton’s method on convex problems , 2005, Math. Program..

[14]  Stephen J. Wright,et al.  Global optimization in protein docking using clustering, underestimation and semidefinite programming , 2007, Optim. Methods Softw..

[15]  Zhiyou Wu,et al.  Global Optimality Conditions for Some Classes of Optimization Problems , 2010 .

[16]  A. Rubinov.  Abstract Convexity and Global Optimization , 2000 .

[17]  H. Tuy.  Convex analysis and global optimization , 1998 .

[18]  Ioannis Ch. Paschalidis,et al.  SDU: A Semidefinite Programming-Based Underestimation Method for Stochastic Global Optimization in Protein Docking , 2007, IEEE Transactions on Automatic Control.

[20]  J. Ben Rosen,et al.  Nonconvex Piecewise-Quadratic Underestimation for Global Minimization , 2006, J. Glob. Optim..

[21]  James Demmel,et al.  Minimizing Polynomials via Sum of Squares over the Gradient Ideal , 2004, Math. Program..

[22]  Jean-Baptiste Hiriart-Urruty,et al.  Global Optimality Conditions in Maximizing a Convex Quadratic Function under Convex Quadratic Constraints , 2001, J. Glob. Optim..

[23]  Yanjun Wang,et al.  Global optimality conditions for cubic minimization problem with box or binary constraints , 2010, J. Glob. Optim..

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

[25]  Graziano Chesi,et al.  LMI Techniques for Optimization Over Polynomials in Control: A Survey , 2010, IEEE Transactions on Automatic Control.

[26]  Zhi-You Wu,et al.  Conditions for Global Optimality of Quadratic Minimization Problems with LMI Constraints , 2007, Asia Pac. J. Oper. Res..

[27]  Marc Teboulle,et al.  Global Optimality Conditions for Quadratic Optimization Problems with Binary Constraints , 2000, SIAM J. Optim..

[28]  M. Pinar,et al.  Sufficient Global Optimality Conditions for Bivalent Quadratic Optimization , 2004 .

[29]  V. Jeyakumar,et al.  New Sufficiency for Global Optimality and Duality of Mathematical Programming Problems via Underestimators , 2009 .

[30]  N. Q. Huy,et al.  Global minimization of difference of quadratic and convex functions over box or binary constraints , 2008, Optim. Lett..

[31]  Didier Henrion,et al.  GloptiPoly: Global optimization over polynomials with Matlab and SeDuMi , 2003, TOMS.

[32]  C. Lin,et al.  Formulation and optimization of cubic polynomial joint trajectories for industrial robots , 1983 .

[33]  Vaithilingam Jeyakumar,et al.  Sufficient Conditions for Global Optimality of Bivalent Nonconvex Quadratic Programs with Inequality Constraints , 2007 .

[34]  R. Horst,et al.  Global Optimization: Deterministic Approaches , 1992 .

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

[36]  Christodoulos A. Floudas,et al.  A New Class of Improved Convex Underestimators for Twice Continuously Differentiable Constrained NLPs , 2004, J. Glob. Optim..