Cross-Hill: A heuristic method for global optimization

The heuristic Cross-Hill method proposed by Qi et?al. (2009) 14 was recently extended from finding the Z-eigenvalues of tensors to quantum separation problem by Han and Qi (2013) 5. In this paper, we show that it can be extended to solve general global optimization problems. The heuristic Cross-Hill method is a combination of a local optimization method and a global optimization method with lower dimension. At each iteration, it first uses the local optimization method to find a local solution. Then, using this point and an arbitrary orthogonal vector, it solves a two-dimensional optimization problem to find a better solution than that the local approach was able to find. Preliminary experimental results are very encouraging.

[1]  Phillip A. Regalia,et al.  On the Best Rank-1 Approximation of Higher-Order Supersymmetric Tensors , 2001, SIAM J. Matrix Anal. Appl..

[2]  Ya-Xiang Yuan,et al.  A Cone Constrained Convex Program: Structure and Algorithms , 2013 .

[3]  Jean B. Lasserre,et al.  Global Optimization with Polynomials and the Problem of Moments , 2000, SIAM J. Optim..

[4]  Liqun Qi,et al.  Rank and eigenvalues of a supersymmetric tensor, the multivariate homogeneous polynomial and the algebraic hypersurface it defines , 2006, J. Symb. Comput..

[5]  Paul H. Calamai,et al.  Projected gradient methods for linearly constrained problems , 1987, Math. Program..

[6]  Pablo A. Parrilo,et al.  Minimizing Polynomial Functions , 2001, Algorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science.

[7]  Neculai Andrei,et al.  An Unconstrained Optimization Test Functions Collection , 2008 .

[8]  Gary S. Tyson,et al.  Evaluating Heuristic Optimization Phase Order Search Algorithms , 2007, International Symposium on Code Generation and Optimization (CGO'07).

[9]  Fei Wang,et al.  Z-eigenvalue methods for a global polynomial optimization problem , 2009, Math. Program..

[10]  Ya-Xiang Yuan,et al.  Optimization Theory and Methods: Nonlinear Programming , 2010 .

[11]  Joos Vandewalle,et al.  On the Best Rank-1 and Rank-(R1 , R2, ... , RN) Approximation of Higher-Order Tensors , 2000, SIAM J. Matrix Anal. Appl..

[12]  Liqun Qi,et al.  Eigenvalues of a real supersymmetric tensor , 2005, J. Symb. Comput..

[13]  Jiawang Nie.  Sum of squares methods for minimizing polynomial forms over spheres and hypersurfaces , 2012 .

[14]  Liqun Qi,et al.  A successive approximation method for quantum separability , 2013 .

[15]  Jorge J. Moré,et al.  Testing Unconstrained Optimization Software , 1981, TOMS.

[16]  Pablo A. Parrilo,et al.  Semidefinite programming relaxations for semialgebraic problems , 2003, Math. Program..