Efficient First-Order Methods for Linear Programming and Semidefinite Programming

Abstract

We present a simple transformation of any linear program or semidefinite program into an equivalent convex optimization problem whose only constraints are linear equations. The objective function is defined on the whole space, making virtually all subgradient methods be immediately applicable. We observe, moreover, that the objective function is naturally smoothed, thereby allowing most first-order methods to be applied. We develop complexity bounds in the unsmoothed case for a particular subgradient method, and in the smoothed case for Nesterov's original optimal first-order method for smooth functions. We achieve the desired bounds on the number of iterations, $O(1/{\epsilon }^2)$ and $O(1/\epsilon )$, respectively. Perhaps most surprising is that the transformation from a linear program or a semidefinite program is simple and so is the basic theory, and yet the approach has been overlooked until now, a blind spot. Once the transformation is realized, the remaining effort in establishing complexity bounds is mainly straightforward, by making use of various works of Nesterov.

Modification made on 10/06/14: Subgradient Method (page 7) was incorrectly specified by giving the last iterate it computes as output, when instead the output should be the iterate with best objective value (as is assumed in the subsequent proofs). The specification of Subgradient Method has now been corrected. No other changes have been made

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