89 research outputs found
New variants of bundle methods
An updated version of this paper has appeared in Mathematical Programming, no 69, (1995), pp. 111-147, DOI 10.1007/BF01585555Résumé disponible dans le fichier PD
Fixed points in the family of convex representations of a maximal monotone operator
Any maximal monotone operator can be characterized by a convex function. The
family of such convex functions is invariant under a transformation connected
with the Fenchel-Legendre conjugation. We prove that there exist a convex
representation of the operator which is a fixed point of this conjugation.Comment: 13 pages, updated references. Submited in July 2002 to Proc. AM
Bundle-based pruning in the max-plus curse of dimensionality free method
Recently a new class of techniques termed the max-plus curse of
dimensionality-free methods have been developed to solve nonlinear optimal
control problems. In these methods the discretization in state space is avoided
by using a max-plus basis expansion of the value function. This requires
storing only the coefficients of the basis functions used for representation.
However, the number of basis functions grows exponentially with respect to the
number of time steps of propagation to the time horizon of the control problem.
This so called "curse of complexity" can be managed by applying a pruning
procedure which selects the subset of basis functions that contribute most to
the approximation of the value function. The pruning procedures described thus
far in the literature rely on the solution of a sequence of high dimensional
optimization problems which can become computationally expensive.
In this paper we show that if the max-plus basis functions are linear and the
region of interest in state space is convex, the pruning problem can be
efficiently solved by the bundle method. This approach combining the bundle
method and semidefinite formulations is applied to the quantum gate synthesis
problem, in which the state space is the special unitary group (which is
non-convex). This is based on the observation that the convexification of the
unitary group leads to an exact relaxation. The results are studied and
validated via examples
Approximate level method
In this paper we propose and analyze a variant of the level method [4], which is an algorithm for minimizing nonsmooth convex functions. The main work per iteration is spent on 1) minimizing a piecewise-linear model of the objective function and on 2) projecting onto the intersection of the feasible region and a polyhedron arising as a level set of the model. We show that by replacing exact computations in both cases by approximate computations, in relative scale, the theoretical iteration complexity increases only by the factor of four. This means that while spending less work on the subproblems, we are able to retain the good theoretical properties of the level method.evel method, approximate projections in relative scale, nonsmooth convex optimization, sensitivity analysis, large-scale optimization.
An optimal subgradient algorithm for large-scale convex optimization in simple domains
This paper shows that the optimal subgradient algorithm, OSGA, proposed in
\cite{NeuO} can be used for solving structured large-scale convex constrained
optimization problems. Only first-order information is required, and the
optimal complexity bounds for both smooth and nonsmooth problems are attained.
More specifically, we consider two classes of problems: (i) a convex objective
with a simple closed convex domain, where the orthogonal projection on this
feasible domain is efficiently available; (ii) a convex objective with a simple
convex functional constraint. If we equip OSGA with an appropriate
prox-function, the OSGA subproblem can be solved either in a closed form or by
a simple iterative scheme, which is especially important for large-scale
problems. We report numerical results for some applications to show the
efficiency of the proposed scheme. A software package implementing OSGA for
above domains is available
Processing second-order stochastic dominance models using cutting-plane representations
This is the post-print version of the Article. The official published version can be accessed from the links below. Copyright @ 2011 Springer-VerlagSecond-order stochastic dominance (SSD) is widely recognised as an important decision criterion in portfolio selection. Unfortunately, stochastic dominance models are known to be very demanding from a computational point of view. In this paper we consider two classes of models which use SSD as a choice criterion. The first, proposed by Dentcheva and Ruszczyński (J Bank Finance 30:433–451, 2006), uses a SSD constraint, which can be expressed as integrated chance constraints (ICCs). The second, proposed by Roman et al. (Math Program, Ser B 108:541–569, 2006) uses SSD through a multi-objective formulation with CVaR objectives. Cutting plane representations and algorithms were proposed by Klein Haneveld and Van der Vlerk (Comput Manage Sci 3:245–269, 2006) for ICCs, and by Künzi-Bay and Mayer (Comput Manage Sci 3:3–27, 2006) for CVaR minimization. These concepts are taken into consideration to propose representations and solution methods for the above class of SSD based models. We describe a cutting plane based solution algorithm and outline implementation details. A computational study is presented, which demonstrates the effectiveness and the scale-up properties of the solution algorithm, as applied to the SSD model of Roman et al. (Math Program, Ser B 108:541–569, 2006).This study was funded by OTKA, Hungarian
National Fund for Scientific Research, project 47340; by Mobile Innovation Centre, Budapest University of Technology, project 2.2; Optirisk Systems, Uxbridge, UK and by BRIEF (Brunel University Research Innovation and Enterprise Fund)
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