71 research outputs found
An Active Set Algorithm for Robust Combinatorial Optimization Based on Separation Oracles
We address combinatorial optimization problems with uncertain coefficients
varying over ellipsoidal uncertainty sets. The robust counterpart of such a
problem can be rewritten as a second-oder cone program (SOCP) with integrality
constraints. We propose a branch-and-bound algorithm where dual bounds are
computed by means of an active set algorithm. The latter is applied to the
Lagrangian dual of the continuous relaxation, where the feasible set of the
combinatorial problem is supposed to be given by a separation oracle. The
method benefits from the closed form solution of the active set subproblems and
from a smart update of pseudo-inverse matrices. We present numerical
experiments on randomly generated instances and on instances from different
combinatorial problems, including the shortest path and the traveling salesman
problem, showing that our new algorithm consistently outperforms the
state-of-the art mixed-integer SOCP solver of Gurobi
Using a Factored Dual in Augmented Lagrangian Methods for Semidefinite Programming
In the context of augmented Lagrangian approaches for solving semidefinite
programming problems, we investigate the possibility of eliminating the
positive semidefinite constraint on the dual matrix by employing a
factorization. Hints on how to deal with the resulting unconstrained
maximization of the augmented Lagrangian are given. We further use the
approximate maximum of the augmented Lagrangian with the aim of improving the
convergence rate of alternating direction augmented Lagrangian frameworks.
Numerical results are reported, showing the benefits of the approach.Comment: 7 page
A Fast Active Set Block Coordinate Descent Algorithm for -regularized least squares
The problem of finding sparse solutions to underdetermined systems of linear
equations arises in several applications (e.g. signal and image processing,
compressive sensing, statistical inference). A standard tool for dealing with
sparse recovery is the -regularized least-squares approach that has
been recently attracting the attention of many researchers. In this paper, we
describe an active set estimate (i.e. an estimate of the indices of the zero
variables in the optimal solution) for the considered problem that tries to
quickly identify as many active variables as possible at a given point, while
guaranteeing that some approximate optimality conditions are satisfied. A
relevant feature of the estimate is that it gives a significant reduction of
the objective function when setting to zero all those variables estimated
active. This enables to easily embed it into a given globally converging
algorithmic framework. In particular, we include our estimate into a block
coordinate descent algorithm for -regularized least squares, analyze
the convergence properties of this new active set method, and prove that its
basic version converges with linear rate. Finally, we report some numerical
results showing the effectiveness of the approach.Comment: 28 pages, 5 figure
A fast branch-and-bound algorithm for non-convex quadratic integer optimization subject to linear constraints using ellipsoidal relaxations
We propose two exact approaches for non-convex quadratic integer minimization subject to linear constraints where lower bounds are computed by considering ellipsoidal relaxations of the feasible set. In the first approach, we intersect the ellipsoids with the feasible linear subspace. In the second approach we penalize exactly the linear constraints. We investigate the connection between both approaches theoretically. Experimental results show that the penalty approach significantly outperforms CPLEX on problems with small or medium size variable domains. © 2015 Elsevier B.V. All rights reserved
An Active-Set Algorithmic Framework for Non-Convex Optimization Problems over the Simplex
In this paper, we describe a new active-set algorithmic framework for
minimizing a non-convex function over the unit simplex. At each iteration, the
method makes use of a rule for identifying active variables (i.e., variables
that are zero at a stationary point) and specific directions (that we name
active-set gradient related directions) satisfying a new "nonorthogonality"
type of condition. We prove global convergence to stationary points when using
an Armijo line search in the given framework. We further describe three
different examples of active-set gradient related directions that guarantee
linear convergence rate (under suitable assumptions). Finally, we report
numerical experiments showing the effectiveness of the approach.Comment: 29 pages, 3 figure
An Active Set Algorithm for Robust Combinatorial Optimization Based on Separation Oracles
We address combinatorial optimization problems with uncertain coefficients varying over ellipsoidal uncertainty sets. The robust counterpart of such a problem can be rewritten as a second-order cone program(SOCP) with integrality constraints. We propose a branch-and-bound algorithm where dual bounds are computed by means of an active set algorithm. The latter is applied to the Lagrangian dual of the continuous relaxation, where the feasible set of the combinatorial problem is supposed to be given by a separation oracle. The method benefits from the closed form solution of the active set subproblems and from a smart update of pseudo-inverse matrices. We present numerical experiments on randomly generated instances and on instances from different combinatorial problems, including the shortest path and the traveling salesman problem, showing that our new algorithm consistently outperforms the state-of-the art mixed-integer SOCP solver of Gurob
Improving ADMMs for Solving Doubly Nonnegative Programs through Dual Factorization
Alternating direction methods of multipliers (ADMMs) are popular approaches
to handle large scale semidefinite programs that gained attention during the
past decade. In this paper, we focus on solving doubly nonnegative programs
(DNN), which are semidefinite programs where the elements of the matrix
variable are constrained to be nonnegative. Starting from two algorithms
already proposed in the literature on conic programming, we introduce two new
ADMMs by employing a factorization of the dual variable.
It is well known that first order methods are not suitable to compute high
precision optimal solutions, however an optimal solution of moderate precision
often suffices to get high quality lower bounds on the primal optimal objective
function value. We present methods to obtain such bounds by either perturbing
the dual objective function value or by constructing a dual feasible solution
from a dual approximate optimal solution. Both procedures can be used as a
post-processing phase in our ADMMs.
Numerical results for DNNs that are relaxations of the stable set problem are
presented. They show the impact of using the factorization of the dual variable
in order to improve the progress towards the optimal solution within an
iteration of the ADMM. This decreases the number of iterations as well as the
CPU time to solve the DNN to a given precision. The experiments also
demonstrate that within a computationally cheap post-processing, we can compute
bounds that are close to the optimal value even if the DNN was solved to
moderate precision only. This makes ADMMs applicable also within a
branch-and-bound algorithm.Comment: 24 pages, 8 figure
Scanning integer points with lex-inequalities: A finite cutting plane algorithm for integer programming with linear objective
We consider the integer points in a unimodular cone K ordered by a
lexicographic rule defined by a lattice basis. To each integer point x in K we
associate a family of inequalities (lex-cuts) that defines the convex hull of
the integer points in K that are not lexicographically smaller than x. The
family of lex-cuts contains the Chvatal-Gomory cuts, but does not contain and
is not contained in the family of split cuts. This provides a finite cutting
plane method to solve the integer program min{cx : x \in S \cap Z^n }, where S
\subset R^n is a compact set and c \in Z^n . We analyze the number of
iterations of our algorithm.Comment: 16 pages, 1 figur
Relax and penalize: a new bilevel approach to mixed-binary hyperparameter optimization
In recent years, bilevel approaches have become very popular to efficiently
estimate high-dimensional hyperparameters of machine learning models. However,
to date, binary parameters are handled by continuous relaxation and rounding
strategies, which could lead to inconsistent solutions. In this context, we
tackle the challenging optimization of mixed-binary hyperparameters by
resorting to an equivalent continuous bilevel reformulation based on an
appropriate penalty term. We propose an algorithmic framework that, under
suitable assumptions, is guaranteed to provide mixed-binary solutions.
Moreover, the generality of the method allows to safely use existing continuous
bilevel solvers within the proposed framework. We evaluate the performance of
our approach for a specific machine learning problem, i.e., the estimation of
the group-sparsity structure in regression problems. Reported results clearly
show that our method outperforms state-of-the-art approaches based on
relaxation and roundin
Analysis of gluten proteins composition during grain filling in two durum wheat cultivars submitted to two water regimes
Durum wheat ( Triticum turgidum L. subsp . durum) is one of the major crops in the Mediterranean basin, where water stress often occurs during grain filling which represents a critical stage for the synthesis and accumulation of storage proteins (gliadins and glutenins). The aim of the study is to evaluate, by two-dimensional gel electrophoresis (2DE SDS-PAGE), the storage proteins composition of two durum wheat cultivars (Ciccio and Svevo) cultivated in a growth chamber under two different water regimes (control and water deficit). At milk stage and physiological maturity, gluten proteins have been extracted and separated by 2DE SDS-PAGE. The analysis of the gels was performed by the software ImageMaster 2D Platinum (Amersham). The results showed differences in protein expression within the different gel regions between water regimes and cultivars; under water deficit the rate of protein accumulation was faster for all the protein regions, either at milk and physiological stage. Protein accumulation within high molecular weight (H) region resulted faster in Ciccio than in Svevo mainly in the control treatment. In the low molecular weight region between 48 and 35 kDa (L 48-35), the cultivar Ciccio showed a higher protein expression than Svevo. Furthermore under water deficit a marked increase in H region volume and a decrease in the L 48-35 region was observed only for Svevo; instead in Ciccio no change was observed showing this cultivar a greater stability on changing water regime. Further studies by the use of mass spectrometry are necessary to identify specific peptides relative to drought stress during grain filling as well as to investigate the relationships with technological quality
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