Rescaling algorithms for linear conic feasibility

We propose simple polynomial-time algorithms for two linear conic feasibility problems. For a matrix A ∈ R m× n, the kernel problem requires a positive vector in the kernel of A, and the image problem requires a positive vector in the image of A T. Both algorithms iterate between simple first-order steps and rescaling steps. These rescalings improve natural geometric potentials. If Goffin's condition measure ρ A is negative, then the kernel problem is feasible, and the worst-case complexity of the kernel algorithm is O((m 3n + mn 2)log|ρ A| −1); if ρ A > 0, then the image problem is feasible, and the image algorithm runs in time O(m 2n 2 log ρ A −1). We also extend the image algorithm to the oracle setting. We address the degenerate case ρA = 0 by extending our algorithms to find maximum support nonnegative vectors in the kernel of A and in the image of A T. In this case, the running time bounds are expressed in the bit-size model of computation: for an input matrix A with integer entries and total encoding length L, the maximum support kernel algorithm runs in time O((m 3n + mn 2)L), whereas the maximum support image algorithm runs in time O(m 2n 2L). The standard linear programming feasibility problem can be easily reduced to either maximum support problems, yielding polynomial-time algorithms for linear programming

A simple method for convex optimization in the oracle model

We give a simple and natural method for computing approximately optimal solutions for minimizing a convex function f over a convex set K given by a separation oracle. Our method utilizes the Frank–Wolfe algorithm over the cone of valid inequalities of K and subgradients of f. Under the assumption that f is L-Lipschitz and that K contains a ball of radius r and is contained inside the origin centered ball of radius R, using O((RL)2ε2·R2r2) iterations and calls to the oracle, our main method outputs a point x∈ K satisfying f(x) ≤ ε+ min z∈Kf(z) . Our algorithm is easy to implement, and we believe it can serve as a useful alternative to existing cutting plane methods. As evidence towards this, we show that it compares favorably in terms of iteration counts to the standard LP based cutting plane method and the analytic center cutting plane method, on a testbed of combinatorial, semidefinite and machine learning instances

Combinatorial Optimization (hybrid meeting)

Combinatorial Optimization deals with optimization problems defined on combinatorial structures such as graphs and networks. Motivated by diverse practical problem setups, the topic has developed into a rich mathematical discipline with many connections to other fields of Mathematics (such as, e.g., Combinatorics, Convex Optimization and Geometry, and Real Algebraic Geometry). It also has strong ties to Theoretical Computer Science and Operations Research. A series of Oberwolfach Workshops have been crucial for establishing and developing the field. The workshop we report about was a particularly exciting event - due to the depth of results that were presented, the spectrum of developments that became apparent from the talks, the breadth of the connections to other mathematical fields that were explored, and last but not least because for many of the particiants it was the first opportunity to exchange ideas and to collaborate during an on-site workshop since almost two years