Efficient interior point algorithms for large scale convex optimization problems
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Date
11/12/2023Author
Zanetti, Filippo
Metadata
Abstract
Interior point methods (IPMs) are among the most widely used algorithms for
convex optimization problems. They are applicable to a wide range of problems, including
linear, quadratic, nonlinear, conic and semidefinite programming problems,
requiring a polynomial number of iterations to find an accurate approximation of
the primal-dual solution. The formidable convergence properties of IPMs come
with a fundamental drawback: the numerical linear algebra involved becomes
progressively more and more challenging as the IPM converges towards optimality.
In particular, solving the linear systems to find the Newton directions requires
most of the computational effort of an IPM. Proposed remedies to alleviate
this phenomenon include regularization techniques, predictor-corrector schemes,
purposely developed preconditioners, low-rank update strategies, to mention a
few.
For problems of very large scale, this unpleasant characteristic of IPMs becomes
a more and more problematic feature, since any technique used must be efficient
and scalable in order to maintain acceptable computational requirements. In this
Thesis, we deal with convex linear and quadratic problems of large “dimension”:
we use this term in a broader sense than just a synonym for “size” of the problem.
The instances considered can be either problems with a large number of variables
and/or constraints but with a sparse structure, or problems with a moderate
number of variables and/or constraints but with a dense structure. Both these
type of problems require very efficient strategies to be used during the algorithm,
even though the corresponding difficulties arise for different reasons.
The first application that we consider deals with a moderate size quadratic
problem where the quadratic term is 100% dense; this problem arises from X-ray
tomographic imaging reconstruction, in particular with the goal of separating the
distribution of two materials present in the observed sample. A novel non-convex
regularizer is introduced for this purpose; convexity of the overall problem is
maintained by careful choice of the parameters. We derive a specialized interior
point method for this problem and an appropriate preconditioner for the normal
equations linear system, to be used without ever forming the fully dense matrices
involved.
The next major contribution is related to the issue of efficiently computing
the Newton direction during IPMs. When an iterative method is applied to
solve the linear equation system in IPMs, the attention is usually placed on
accelerating their convergence by designing appropriate preconditioners, but the
linear solver is applied as a black box with a standard termination criterion
which asks for a sufficient reduction of the residual in the linear system. Such an
approach often leads to an unnecessary “over-solving” of linear equations. We
propose new indicators for the early termination of the inner iterations and test
them on a set of large scale quadratic optimization problems. Evidence gathered
from these computational experiments shows that the new technique delivers
significant improvements in terms of inner (linear) iterations and those translate
into significant savings of the IPM solution time.
The last application considered is discrete optimal transport (OT) problems;
these kind of problems give rise to very large linear programs with highly structured
matrices. Solutions of such problems are expected to be sparse, that is only a
small subset of entries in the optimal solution is expected to be nonzero. We derive
an IPM for the standard OT formulation, which exploits a column-generation-like
technique to force all intermediate iterates to be as sparse as possible. We prove
theoretical results about the sparsity pattern of the optimal solution and we
propose to mix iterative and direct linear solvers in an efficient way, to keep
computational time and memory requirement as low as possible. We compare the
proposed method with two state-of-the-art solvers and show that it can compete
with the best network optimization tools in terms of computational time and
memory usage. We perform experiments with problems reaching more than four
billion variables and demonstrate the robustness of the proposed method.
We consider also the optimal transport problem on sparse graphs and present
a primal-dual regularized IPM to solve it. We prove that the introduction of the
regularization allows us to use sparsified versions of the normal equations system
to inexpensively generate inexact IPM directions. The proposed method is shown
to have polynomial complexity and to outperform a very efficient network simplex
implementation, for problems with up to 50 million variables.