10,439 research outputs found
A Parametric Simplex Algorithm for Linear Vector Optimization Problems
In this paper, a parametric simplex algorithm for solving linear vector
optimization problems (LVOPs) is presented. This algorithm can be seen as a
variant of the multi-objective simplex (Evans-Steuer) algorithm [12]. Different
from it, the proposed algorithm works in the parameter space and does not aim
to find the set of all efficient solutions. Instead, it finds a solution in the
sense of Loehne [16], that is, it finds a subset of efficient solutions that
allows to generate the whole frontier. In that sense, it can also be seen as a
generalization of the parametric self-dual simplex algorithm, which originally
is designed for solving single objective linear optimization problems, and is
modified to solve two objective bounded LVOPs with the positive orthant as the
ordering cone in Ruszczynski and Vanderbei [21]. The algorithm proposed here
works for any dimension, any solid pointed polyhedral ordering cone C and for
bounded as well as unbounded problems. Numerical results are provided to
compare the proposed algorithm with an objective space based LVOP algorithm
(Benson algorithm in [13]), that also provides a solution in the sense of [16],
and with Evans-Steuer algorithm [12]. The results show that for non-degenerate
problems the proposed algorithm outperforms Benson algorithm and is on par with
Evan-Steuer algorithm. For highly degenerate problems Benson's algorithm [13]
excels the simplex-type algorithms; however, the parametric simplex algorithm
is for these problems computationally much more efficient than Evans-Steuer
algorithm.Comment: 27 pages, 4 figures, 5 table
Polynomial Optimization with Applications to Stability Analysis and Control - Alternatives to Sum of Squares
In this paper, we explore the merits of various algorithms for polynomial
optimization problems, focusing on alternatives to sum of squares programming.
While we refer to advantages and disadvantages of Quantifier Elimination,
Reformulation Linear Techniques, Blossoming and Groebner basis methods, our
main focus is on algorithms defined by Polya's theorem, Bernstein's theorem and
Handelman's theorem. We first formulate polynomial optimization problems as
verifying the feasibility of semi-algebraic sets. Then, we discuss how Polya's
algorithm, Bernstein's algorithm and Handelman's algorithm reduce the
intractable problem of feasibility of semi-algebraic sets to linear and/or
semi-definite programming. We apply these algorithms to different problems in
robust stability analysis and stability of nonlinear dynamical systems. As one
contribution of this paper, we apply Polya's algorithm to the problem of
H_infinity control of systems with parametric uncertainty. Numerical examples
are provided to compare the accuracy of these algorithms with other polynomial
optimization algorithms in the literature.Comment: AIMS Journal of Discrete and Continuous Dynamical Systems - Series
An Exponential Lower Bound on the Complexity of Regularization Paths
For a variety of regularized optimization problems in machine learning,
algorithms computing the entire solution path have been developed recently.
Most of these methods are quadratic programs that are parameterized by a single
parameter, as for example the Support Vector Machine (SVM). Solution path
algorithms do not only compute the solution for one particular value of the
regularization parameter but the entire path of solutions, making the selection
of an optimal parameter much easier.
It has been assumed that these piecewise linear solution paths have only
linear complexity, i.e. linearly many bends. We prove that for the support
vector machine this complexity can be exponential in the number of training
points in the worst case. More strongly, we construct a single instance of n
input points in d dimensions for an SVM such that at least \Theta(2^{n/2}) =
\Theta(2^d) many distinct subsets of support vectors occur as the
regularization parameter changes.Comment: Journal version, 28 Pages, 5 Figure
Learning by mirror averaging
Given a finite collection of estimators or classifiers, we study the problem
of model selection type aggregation, that is, we construct a new estimator or
classifier, called aggregate, which is nearly as good as the best among them
with respect to a given risk criterion. We define our aggregate by a simple
recursive procedure which solves an auxiliary stochastic linear programming
problem related to the original nonlinear one and constitutes a special case of
the mirror averaging algorithm. We show that the aggregate satisfies sharp
oracle inequalities under some general assumptions. The results are applied to
several problems including regression, classification and density estimation.Comment: Published in at http://dx.doi.org/10.1214/07-AOS546 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Lagrangean decomposition for large-scale two-stage stochastic mixed 0-1 problems
In this paper we study solution methods for solving the dual problem corresponding to the Lagrangean Decomposition of two stage stochastic mixed 0-1 models. We represent the two stage stochastic mixed 0-1 problem by a splitting variable representation of the deterministic equivalent model, where 0-1 and continuous variables appear at any stage. Lagrangean Decomposition is proposed for satisfying both the integrality constraints for the 0-1 variables and the non-anticipativity constraints. We compare the performance of four iterative algorithms based on dual Lagrangean Decomposition schemes, as the Subgradient method, the Volume algorithm, the Progressive Hedging algorithm and the Dynamic Constrained Cutting Plane scheme. We test the conditions and properties of convergence for medium and large-scale dimension stochastic problems. Computational results are reported.Progressive Hedging algorithm, volume algorithm, Lagrangean decomposition, subgradient method
OSQP: An Operator Splitting Solver for Quadratic Programs
We present a general-purpose solver for convex quadratic programs based on
the alternating direction method of multipliers, employing a novel operator
splitting technique that requires the solution of a quasi-definite linear
system with the same coefficient matrix at almost every iteration. Our
algorithm is very robust, placing no requirements on the problem data such as
positive definiteness of the objective function or linear independence of the
constraint functions. It can be configured to be division-free once an initial
matrix factorization is carried out, making it suitable for real-time
applications in embedded systems. In addition, our technique is the first
operator splitting method for quadratic programs able to reliably detect primal
and dual infeasible problems from the algorithm iterates. The method also
supports factorization caching and warm starting, making it particularly
efficient when solving parametrized problems arising in finance, control, and
machine learning. Our open-source C implementation OSQP has a small footprint,
is library-free, and has been extensively tested on many problem instances from
a wide variety of application areas. It is typically ten times faster than
competing interior-point methods, and sometimes much more when factorization
caching or warm start is used. OSQP has already shown a large impact with tens
of thousands of users both in academia and in large corporations
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