On The Geometric Analysis of A Quartic-quadratic Optimization Problem under A Spherical Constraint

Haixiang Zhang, Andre Milzarek, Zaiwen Wen, and Wotao Yin

Submitted:

Overview

This paper considers the problem of solving a special quartic-quadratic optimization problem with a single sphere constraint, specifically, finding a global and local minimizer of frac{1}{2}z^*Az + frac{beta}{2}sum_{k=1}^n|z_k|^4 such that |z|_2=1. This problem spans multiple domains including quantum mechanics and chemistry sciences and we investigate the geometric properties of this optimization problem. Fourth-order optimality conditions are derived for characterizing local and global minima. When the matrix in the quadratic term is diagonal, the problem has no spurious local minima and global solutions can be represented explicitly and calculated in O(nlog n) operations. When the matrix is a rank one matrix, the global minima of the problem are unique under certain phase shift schemes. The strict-saddle property, which can imply polynomial time convergence of second-order-type algorithms, is established when the coefficient beta of the quartic term is either at least O(n^{3/2}) or not larger than O(1). Finally, the Kurdyka-Lojasiewicz exponent of quartic-quadratic problem is estimated and it is shown that the exponent is 1/4 for a broad class of stationary points.

The plot below depicts the landscape of the objective function with fixed matrix A and different values of beta.

  • Red dots are saddle points.

  • Empty and filled diamond are local and global minima, respectively.

  • Empty and filled squares are local and global maxima, respectively.

 

Citation

H. Zhang, A. Milzarek, Z. Wen, and W. Yin, On the geometric analysis of a quartic-quadratic optimization problem under a spherical constraint, arXiv:1908.00745, 2019.


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