90 research outputs found
An additive subfamily of enlargements of a maximally monotone operator
We introduce a subfamily of additive enlargements of a maximally monotone
operator. Our definition is inspired by the early work of Simon Fitzpatrick.
These enlargements constitute a subfamily of the family of enlargements
introduced by Svaiter. When the operator under consideration is the
subdifferential of a convex lower semicontinuous proper function, we prove that
some members of the subfamily are smaller than the classical
-subdifferential enlargement widely used in convex analysis. We also
recover the epsilon-subdifferential within the subfamily. Since they are all
additive, the enlargements in our subfamily can be seen as structurally closer
to the -subdifferential enlargement
Non-smooth optimization methods for computation of the conditional value-at-risk and portfolio optimization
We examine numerical performance of various methods of calculation of the Conditional Value-at-risk (CVaR), and portfolio optimization with respect to this risk measure. We concentrate on the method proposed by Rockafellar and Uryasev in (Rockafellar, R.T. and Uryasev, S., 2000, Optimization of conditional value-at-risk. Journal of Risk, 2, 21-41), which converts this problem to that of convex optimization. We compare the use of linear programming techniques against a non-smooth optimization method of the discrete gradient, and establish the supremacy of the latter. We show that non-smooth optimization can be used efficiently for large portfolio optimization, and also examine parallel execution of this method on computer clusters.<br /
Approximation of the critical buckling factor for composite panels
This article is concerned with the approximation of the critical buckling factor for thin composite plates. A new method to improve the approximation of this critical factor is applied based on its behavior with respect to lamination parameters and loading conditions. This method allows accurate approximation of the critical buckling factor for non-orthotropic laminates under complex combined loadings (including shear loading). The influence of the stacking sequence and loading conditions is extensively studied as well as properties of the critical buckling factor behavior (e.g concavity over tensor D or out-of-plane lamination parameters). Moreover, the critical buckling factor is numerically shown to be piecewise linear for orthotropic laminates under combined loading whenever shear remains low and it is also shown to be piecewise continuous in the general case. Based on the numerically observed behavior, a new scheme for the approximation is applied that separates each buckling mode and builds linear, polynomial or rational regressions for each mode. Results of this approach and applications to structural optimization are presented
Large-scale optimization with the primal-dual column generation method
The primal-dual column generation method (PDCGM) is a general-purpose column
generation technique that relies on the primal-dual interior point method to
solve the restricted master problems. The use of this interior point method
variant allows to obtain suboptimal and well-centered dual solutions which
naturally stabilizes the column generation. As recently presented in the
literature, reductions in the number of calls to the oracle and in the CPU
times are typically observed when compared to the standard column generation,
which relies on extreme optimal dual solutions. However, these results are
based on relatively small problems obtained from linear relaxations of
combinatorial applications. In this paper, we investigate the behaviour of the
PDCGM in a broader context, namely when solving large-scale convex optimization
problems. We have selected applications that arise in important real-life
contexts such as data analysis (multiple kernel learning problem),
decision-making under uncertainty (two-stage stochastic programming problems)
and telecommunication and transportation networks (multicommodity network flow
problem). In the numerical experiments, we use publicly available benchmark
instances to compare the performance of the PDCGM against recent results for
different methods presented in the literature, which were the best available
results to date. The analysis of these results suggests that the PDCGM offers
an attractive alternative over specialized methods since it remains competitive
in terms of number of iterations and CPU times even for large-scale
optimization problems.Comment: 28 pages, 1 figure, minor revision, scaled CPU time
Learning in games with continuous action spaces and unknown payoff functions
International audienc
Solving Unit Commitment Problems in Power Production Planning
Unit commitment in production planning of power systems involves dynamic, mixed-integer programming problems. We show that specialized bundle methods applied to Lagrangian relaxations provide not only lower bounds on the optimal value (as do subgradient methods), but also certain relaxed primal solutions. These solutions can be used for constructing good primal feasible solutions. We present computational experience for large-scale examples. 1 Introduction The unit commitment (UC) problem in the daily operation of power systems determines on/off schedules and power outputs of the generators, so as to minimize the system operating cost over a planning horizon of 24 to 168 hours. It is a large-scale mixed integer programming problem. Many solution methodologies [ShF94] have been proposed for the UC problem. Among the most promising is Lagrangian relaxation [MuK77, MoR94, LPRS96], in which a dual function is formed by minimizing the cost augmented with coupling constraints weighted by La..
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