An Augmented Primal-Dual Method for Linear Conic Programs

Abstract

Abstract. We propose a new iterative approach for solving linear programs over convex cones. Assuming that Slaters condition is satisfied, the conic problem is transformed to the minimization of a convex differentiable function. This "agumented primal-dual function" or "apd-function" is restricted to an affine set in the primal-dual space. The evaluation of the function and its derivative is cheap if the projection of a given point onto the cone can be computed cheaply, and if the projection of a given point onto the affine subspace defining the primal problem can be computed cheaply. For the special case of a semidefinite program, a certain regularization of the apd-function is analyzed. Numerical examples minimizing the apd-function with a conjugate gradient method illustrate the potential of the approach.

Projection Methods in Conic Optimization

“…The developments of this section share similar properties with other augmented Lagrangian-type approaches for conic programming, among them: a primal augmented Lagrangian in [BV06], a primal-dual augmented Lagrangian in [JR08] and a penalized augmented Lagrangian in [KS07].…”

Projection methods for conic feasibility problems: applications to polynomial sum-of-squares decompositions

“…[50] and references therein). Efficiently solving conic feasibility can also have applications in conic programming itself, for example, for finding starting points or, following [28], for solving linear conic programming problems as conic feasibility problems after formulation of the optimality conditions.…”

On the computation of C* certificates

“…The augmented primal dual (apd) method of [9,13] is applied to conic programs in a certain standard form due to Nesterov and Nemirovskii [12],…”

“…The main contributions of the present paper are the introduction of a regularization step, the discussion of its implementation, and finally, some encouraging numerical examples. The numerical examples are based on the recent "augmented primal-dual" (apd) method in [9], exploiting the structure of the large scale sub-problems by a first order method for conic optimization that quickly generates approximate solutions but that is less suitable for high accuracy solutions.…”