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We give a rigorous complexity analysis of the simulated annealing algorithmby Kalai and Vempala [Math of OR 31.2 (2006): 253-266] using the type oftemperature update suggested by Abernethy and Hazan [arXiv 1507.02528v2, 2015].The algorithm only assumes a membership oracle of the feasible set, and weprove that it returns a solution in polynomial time which is near-optimal withhigh probability. Moreover, we propose a number of modifications to improve thepractical performance of this method, and present some numerical results fortest problems from copositive programming.

Simulated annealing with hit-and-run for convex optimization: rigorous complexity analysis and practical perspectives for copositive programming