Self-Scaled Barriers and Interior-Point Methods for Convex Programming

Abstract

This paper provides a theoretical foundation for efficient interior-point algorithms for convex programming problems expressed in conic form, when the cone and its associated barrier are self-scaled. For such problems we devise long-step and symmetric primal-dual methods. Because of the special properties of these cones and barriers, our algorithms can take steps that go typically a large fraction of the way to the boundary of the feasible region, rather than being confined to a ball of unit radius in the local norm defined by the Hessian of the barrier.

A homogeneous interior-point algorithm for nonsymmetric convex conic optimization

“…Thus no particular emphasis is put on either the primal or the dual problem, which is a desirable feature of an algorithm. If K is self-scaled, a symmetric u (μ) can be computed using a scaling point [17,18]. If the cone is not self-scaled (nonsymmetric), a symmetric u (μ) can be computed by using both the Hessian of the primal and the dual barrier.…”

“…If K admits a self-scaled barrier function F: K • → R, problems of the type (pd) are efficiently solvable using long-step symmetric primal-dual ipms [17,18]. The practical efficiency of these algorithms has been widely verified, see e.g.…”

Primal-dual algorithms and infinite-dimensional Jordan algebras of finite rank

“…Let us illustrate this point by considering a long-step primal-dual algorithm based on the Nesterov-Todd direction [13].…”

“…The class of scaling-invariant "descent" directions is obtained by solving the following system of linear equations. Given (z, w) ∈ Ω×Ω and g ∈ GL(Ω) (see (13)), observe first of all that g −T ∈ GL(Ω), sinceΩ * =Ω. The system of linear equations has the form:…”

Nonlinear semidefinite programming: sensitivity, convergence, and an application in passive reduced-order modeling

“…The computation of the search direction is often carried out by forming a Schur complement matrix. For the HKM search direction [16,19,22] and the NT search direction [24,25], this matrix is positive definite, and this is the essential property that guarantees a descent property of the search step. On the other hand, for the AHO search direction [2], the Schur complement matrix is not symmetric.…”