Efficient Convex Optimization with Oracles

Minimizing a convex function over a convex set is a basic algorithmic problem. We give a simple algorithm for the general setting in which the function is given by an evaluation oracle and the set by a membership oracle. The algorithm takes \(\widetilde{O}(n^{2})\) oracle calls and \(\widetilde{O}(n^{3})\) additional arithmetic operations. This results in more efficient reductions among the five basic oracles for convex sets and functions defined by Grotschel, Lovasz and Schrijver (Algorithms Comb 2, (1988), [5]).

[1]  Martin Grötschel,et al.  Geometric Algorithms and Combinatorial Optimization , 1988, Algorithms and Combinatorics.

[2]  V. Protasov.  Algorithms for approximate calculation of the minimum of a convex function from its values , 1996 .

[3]  Pravin M. Vaidya,et al.  A new algorithm for minimizing convex functions over convex sets , 1996, Math. Program..

[4]  Santosh S. Vempala,et al.  Solving convex programs by random walks , 2004, JACM.

[5]  Santosh S. Vempala,et al.  Simulated Annealing for Convex Optimization , 2004 .

[6]  Santosh S. Vempala,et al.  Fast Algorithms for Logconcave Functions: Sampling, Rounding, Integration and Optimization , 2006, 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06).

[7]  Shang-Hua Teng,et al.  Electrical flows, laplacian systems, and faster approximation of maximum flow in undirected graphs , 2010, STOC '11.

[8]  Aleksander Madry,et al.  Navigating Central Path with Electrical Flows: From Flows to Matchings, and Back , 2013, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science.

[9]  Satish Rao,et al.  A new approach to computing maximum flows using electrical flows , 2013, STOC '13.

[10]  Jonah Sherman,et al.  Nearly Maximum Flows in Nearly Linear Time , 2013, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science.

[11]  S. Steinerberger.  Sharp L1-Poincaré inequalities correspond to optimal hypersurface cuts , 2013, 1309.6211.

[12]  Yin Tat Lee,et al.  An Almost-Linear-Time Algorithm for Approximate Max Flow in Undirected Graphs, and its Multicommodity Generalizations , 2013, SODA.

[13]  Yin Tat Lee,et al.  Path Finding Methods for Linear Programming: Solving Linear Programs in Õ(vrank) Iterations and Faster Algorithms for Maximum Flow , 2014, 2014 IEEE 55th Annual Symposium on Foundations of Computer Science.

[14]  Yin Tat Lee,et al.  A Faster Cutting Plane Method and its Implications for Combinatorial and Convex Optimization , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[15]  Yin Tat Lee,et al.  Efficient Inverse Maintenance and Faster Algorithms for Linear Programming , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[16]  Elad Hazan,et al.  Faster Convex Optimization: Simulated Annealing with an Efficient Universal Barrier , 2015, ICML.

[17]  Sébastien Bubeck,et al.  Multi-scale exploration of convex functions and bandit convex optimization , 2015, COLT.

[18]  Yurii Nesterov,et al.  Random Gradient-Free Minimization of Convex Functions , 2015, Foundations of Computational Mathematics.

[19]  Jonah Sherman,et al.  Area-convexity, l∞ regularization, and undirected multicommodity flow , 2017, STOC.