161 research outputs found
A semismooth newton method for the nearest Euclidean distance matrix problem
The Nearest Euclidean distance matrix problem (NEDM) is a fundamentalcomputational problem in applications such asmultidimensional scaling and molecularconformation from nuclear magnetic resonance data in computational chemistry.Especially in the latter application, the problem is often large scale with the number ofatoms ranging from a few hundreds to a few thousands.In this paper, we introduce asemismooth Newton method that solves the dual problem of (NEDM). We prove that themethod is quadratically convergent.We then present an application of the Newton method to NEDM with -weights.We demonstrate the superior performance of the Newton method over existing methodsincluding the latest quadratic semi-definite programming solver.This research also opens a new avenue towards efficient solution methods for the molecularembedding problem
Forward-backward truncated Newton methods for convex composite optimization
This paper proposes two proximal Newton-CG methods for convex nonsmooth
optimization problems in composite form. The algorithms are based on a a
reformulation of the original nonsmooth problem as the unconstrained
minimization of a continuously differentiable function, namely the
forward-backward envelope (FBE). The first algorithm is based on a standard
line search strategy, whereas the second one combines the global efficiency
estimates of the corresponding first-order methods, while achieving fast
asymptotic convergence rates. Furthermore, they are computationally attractive
since each Newton iteration requires the approximate solution of a linear
system of usually small dimension
A New Inexact Non-Interior Continuation Algorithm for Second-Order Cone Programming
Second-order cone programming has received considerable attention in the past decades because of its wide range of applications. Non-interior continuation method is one of the most popular and efficient methods for solving second-order cone programming partially due to its superior numerical performances. In this paper, a new smoothing form of the well-known Fischer-Burmeister function is given. Based on the new smoothing function, an inexact non-interior continuation algorithm is proposed. Attractively, the new algorithm can start from an arbitrary point, and it solves only one system of linear equations inexactly and performs only one line search at each iteration. Moreover, under a mild assumption, the new algorithm has a globally linear and locally Q-quadratical convergence. Finally, some preliminary numerical results are reported which show the effectiveness of the presented algorithm
A squared smoothing Newton method for nonsmooth matrix equations and its applications in semidefinite optimization problems
10.1137/S1052623400379620SIAM Journal on Optimization143783-80
Variational Properties of Decomposable Functions Part II: Strong Second-Order Theory
Local superlinear convergence of the semismooth Newton method usually
requires the uniform invertibility of the generalized Jacobian matrix, e.g.
BD-regularity or CD-regularity. For several types of nonlinear programming and
composite-type optimization problems -- for which the generalized Jacobian of
the stationary equation can be calculated explicitly -- this is characterized
by the strong second-order sufficient condition. However, general
characterizations are still not well understood. In this paper, we propose a
strong second-order sufficient condition (SSOSC) for composite problems whose
nonsmooth part has a generalized conic-quadratic second subderivative. We then
discuss the relationship between the SSOSC and another second order-type
condition that involves the generalized Jacobians of the normal map. In
particular, these two conditions are equivalent under certain structural
assumptions on the generalized Jacobian matrix of the proximity operator. Next,
we verify these structural assumptions for -strictly decomposable
functions via analyzing their second-order variational properties under
additional geometric assumptions on the support set of the decomposition pair.
Finally, we show that the SSOSC is further equivalent to the strong metric
regularity condition of the subdifferential, the normal map, and the natural
residual. Counterexamples illustrate the necessity of our assumptions.Comment: 28 pages; preliminary draf
An efficient algorithm for the norm based metric nearness problem
Given a dissimilarity matrix, the metric nearness problem is to find the
nearest matrix of distances that satisfy the triangle inequalities. This
problem has wide applications, such as sensor networks, image processing, and
so on. But it is of great challenge even to obtain a moderately accurate
solution due to the metric constraints and the nonsmooth objective
function which is usually a weighted norm based distance. In this
paper, we propose a delayed constraint generation method with each subproblem
solved by the semismooth Newton based proximal augmented Lagrangian method
(PALM) for the metric nearness problem. Due to the high memory requirement for
the storage of the matrix related to the metric constraints, we take advantage
of the special structure of the matrix and do not need to store the
corresponding constraint matrix. A pleasing aspect of our algorithm is that we
can solve these problems involving up to variables and
constraints. Numerical experiments demonstrate the efficiency of our algorithm.
In theory, firstly, under a mild condition, we establish a primal-dual error
bound condition which is very essential for the analysis of local convergence
rate of PALM. Secondly, we prove the equivalence between the dual nondegeneracy
condition and nonsingularity of the generalized Jacobian for the inner
subproblem of PALM. Thirdly, when or
, without the strict complementarity condition, we also
prove the equivalence between the the dual nondegeneracy condition and the
uniqueness of the primal solution
Linear complementarity problems on extended second order cones
In this paper, we study the linear complementarity problems on extended
second order cones. We convert a linear complementarity problem on an extended
second order cone into a mixed complementarity problem on the non-negative
orthant. We state necessary and sufficient conditions for a point to be a
solution of the converted problem. We also present solution strategies for this
problem, such as the Newton method and Levenberg-Marquardt algorithm. Finally,
we present some numerical examples
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