125 research outputs found
An Inexact Augmented Lagrangian Method for Second-order Cone Programming with Applications
In this paper, we adopt the augmented Lagrangian method (ALM) to solve convex
quadratic second-order cone programming problems (SOCPs). Fruitful results on
the efficiency of the ALM have been established in the literature. Recently, it
has been shown in [Cui, Sun, and Toh, {\em Math. Program.}, 178 (2019), pp.
381--415] that if the quadratic growth condition holds at an optimal solution
for the dual problem, then the KKT residual converges to zero R-superlinearly
when the ALM is applied to the primal problem. Moreover, Cui, Ding, and Zhao
[{\em SIAM J. Optim.}, 27 (2017), pp. 2332-2355] provided sufficient conditions
for the quadratic growth condition to hold under the metric subregularity and
bounded linear regularity conditions for solving composite matrix optimization
problems involving spectral functions. Here, we adopt these recent ideas to
analyze the convergence properties of the ALM when applied to SOCPs. To the
best of our knowledge, no similar work has been done for SOCPs so far. In our
paper, we first provide sufficient conditions to ensure the quadratic growth
condition for SOCPs. With these elegant theoretical guarantees, we then design
an SOCP solver and apply it to solve various classes of SOCPs, such as minimal
enclosing ball problems, classical trust-region subproblems, square-root Lasso
problems, and DIMACS Challenge problems. Numerical results show that the
proposed ALM based solver is efficient and robust compared to the existing
highly developed solvers, such as Mosek and SDPT3.Comment: 25 pages, 0 figur
Projection methods in conic optimization
There exist efficient algorithms to project a point onto the intersection of
a convex cone and an affine subspace. Those conic projections are in turn the
work-horse of a range of algorithms in conic optimization, having a variety of
applications in science, finance and engineering. This chapter reviews some of
these algorithms, emphasizing the so-called regularization algorithms for
linear conic optimization, and applications in polynomial optimization. This is
a presentation of the material of several recent research articles; we aim here
at clarifying the ideas, presenting them in a general framework, and pointing
out important techniques
Development of a nonlinear equations solver with superlinear convergence at regular singularities
In dieser Arbeit präsentieren wir eine neue Art von Newton-Verfahren mit Liniensuche, basierend auf Interpolation im Bildbereich nach Wedin et al. [LW84]. Von dem resultierenden stabilisierten Newton-Algorithmus wird theoretisch und praktisch gezeigt, dass er effizient ist im Falle von nichtsingulären Lösungen. Darüber hinaus wird beobachtet, dass er eine superlineare Rate von Konvergenz bei einfachen Singularitäten erhält. Hingegen ist vom Newton-Verfahren ohne Liniensuche bekannt, dass es nur linear von fast allen Punkten in der Nähe einer singulären Lösung konvergiert. In Hinsicht auf Anwendungen auf Komplementaritätsprobleme betrachten wir auch Systeme, deren Jacobimatrix nicht differenzierbar sondern nur semismooth ist. Auch hier erreicht unser stabilisiertes und beschleunigtes Newton- Verfahren Superlinearität bei einfachen Singularitäten.In this thesis we present a new type of line-search for Newton’s method, based on range space interpolation as suggested by Wedin et al. [LW84]. The resulting stabilized Newton algorithm is theoretically and practically shown to be efficient in the case of nonsingular roots. Moreover it is observed that it maintains a superlinear rate of convergence at simple singularities. Whereas Newton’s method without line-search is known to converge only linearly from almost all points near the singular root. In view of applications to complementarity problems we also consider systems, whose Jacobian is not differentiable but only semismooth. Again, our stabilized and accelerated Newton’s method achieves superlinearity at simple singularities
A Smoothing Newton-BICGStab Method for Least Squares Matrix Nuclear Norm Problems
Master'sMASTER OF SCIENC
A trust region-type normal map-based semismooth Newton method for nonsmooth nonconvex composite optimization
We propose a novel trust region method for solving a class of nonsmooth and
nonconvex composite-type optimization problems. The approach embeds inexact
semismooth Newton steps for finding zeros of a normal map-based stationarity
measure for the problem in a trust region framework. Based on a new merit
function and acceptance mechanism, global convergence and transition to fast
local q-superlinear convergence are established under standard conditions. In
addition, we verify that the proposed trust region globalization is compatible
with the Kurdyka-{\L}ojasiewicz (KL) inequality yielding finer convergence
results. We further derive new normal map-based representations of the
associated second-order optimality conditions that have direct connections to
the local assumptions required for fast convergence. Finally, we study the
behavior of our algorithm when the Hessian matrix of the smooth part of the
objective function is approximated by BFGS updates. We successfully link the KL
theory, properties of the BFGS approximations, and a Dennis-Mor{\'e}-type
condition to show superlinear convergence of the quasi-Newton version of our
method. Numerical experiments on sparse logistic regression and image
compression illustrate the efficiency of the proposed algorithm.Comment: 56 page
A Generalized Newton Method for Subgradient Systems
This paper proposes and develops a new Newton-type algorithm to solve
subdifferential inclusions defined by subgradients of extended-real-valued
prox-regular functions. The proposed algorithm is formulated in terms of the
second-order subdifferential of such functions that enjoys extensive calculus
rules and can be efficiently computed for broad classes of extended-real-valued
functions. Based on this and on metric regularity and subregularity properties
of subgradient mappings, we establish verifiable conditions ensuring
well-posedness of the proposed algorithm and its local superlinear convergence.
The obtained results are also new for the class of equations defined by
continuously differentiable functions with Lipschitzian derivatives
( functions), which is the underlying case of our
consideration. The developed algorithm for prox-regular functions is formulated
in terms of proximal mappings related to and reduces to Moreau envelopes.
Besides numerous illustrative examples and comparison with known algorithms for
functions and generalized equations, the paper presents
applications of the proposed algorithm to the practically important class of
Lasso problems arising in statistics and machine learning.Comment: 35 page
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