Globally minimizing the sum of a convex-concave fraction and a convex function based on wave-curve bounds

摘要

We consider the problem of minimizing the sum of a convex-concave function and a convex function over a convex set (SFC). It can be reformulated as a univariate minimization problem, where the objective function is evaluated by solving convex optimization. The optimal Lagrangian multipliers of the convex subproblems are used to construct sawtooth curve lower bounds, which play a key role in developing the branch-and-bound algorithm for globally solving (SFC). In this paper, we improve the existing sawtooth-curve bounds to new wave-curve bounds, which are used to develop a more efficient branch-and-bound algorithm. Moreover, we can show that the new algorithm finds an epsilondocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$epsilon $$end{document}-approximate optimal solution in at most O mml:mfenced close=")" open="("1 epsilondocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$Oleft( rac{1}{epsilon }ight) $$end{document} iterations. Numerical results demonstrate the efficiency of our algorithm.