Level Bundle Method Research Articles

Discrete level-bundle method for mean-CVaR portfolio optimization with cardinality constraint

Discrete level-bundle method for mean-CVaR portfolio optimization with cardinality constraint

A new restricted memory level bundle method for constrained convex nonsmooth optimization

A new restricted memory level bundle method for constrained convex nonsmooth optimization

Two-stage and multi-stage decompositions for the medium-term hydrothermal scheduling problem: A computational comparison of solution techniques

In hydro-based power systems, the Generation Scheduling (GS) problem is usually divided into a series of coupled models aiming to obtain an optimal operating policy over a multi-year planning horizon. The medium-term GS (MTGS) problem belongs to these models, linking the long-term GS (LTGS) with the short-term GS (STGS) problem. To carry out a proper coupling, we propose a stochastic MTGS model with a multi-period scenario tree with non-common samples, and hourly discretization in the first week to include the nonconvex thermal unit-commitment constraints. These considerations lead to solving a very challenging large-scale stochastic optimization problem with mixed-binary variables in the first week. In this context, efficient solution techniques must be explored to ensure a suitable trade-off between solution accuracy and computational performance. In this paper, we propose a two-stage decomposition for the MTGS that is solved by the extended level bundle method. Every second-stage subproblem is itself a multi-period stochastic problem involving only continuous variables. Numerical assessments on the large-scale Brazilian MTGS problem indicate that our approach significantly outperforms the Nested decomposition and Stochastic Dual Dynamic Programming (SDDP) algorithm in terms of computational burden, providing CPU time reductions in the order of 19%.

Acceleration techniques for level bundle methods in weakly smooth convex constrained optimization

We develop a unified level-bundle method, called accelerated constrained level-bundle (ACLB) algorithm, for solving constrained convex optimization problems. where the objective and constraint functions can be nonsmooth, weakly smooth, and/or smooth. ACLB employs Nesterov’s accelerated gradient technique, and hence retains the iteration complexity as that of existing bundle-type methods if the objective or one of the constraint functions is nonsmooth. More importantly, ACLB can significantly reduce iteration complexity when the objective and all constraints are (weakly) smooth. In addition, if the objective contains a nonsmooth component which can be written as a specific form of maximum, we show that the iteration complexity of this component can be much lower than that for general nonsmooth objective function. Numerical results demonstrate the effectiveness of the proposed algorithm.

Combining Lagrangian relaxation, benders decomposition, and the level bundle method in the stochastic hydrothermal unit‐commitment problem

Combining Lagrangian relaxation, benders decomposition, and the level bundle method in the stochastic hydrothermal unit‐commitment problem

Asynchronous level bundle methods

In this paper, we consider nonsmooth convex optimization problems with additive structure featuring independent oracles (black-boxes) working in parallel. Existing methods for solving these distributed problems in a general form are synchronous, in the sense that they wait for the responses of all the oracles before performing a new iteration. In this paper, we propose level bundle methods handling asynchronous oracles. These methods require original upper-bounds (using upper-models or scarce coordinations) to deal with asynchronicity. We prove their convergence using variational-analysis techniques and illustrate their practical performance on a Lagrangian decomposition problem.

On level regularization with normal solutions in decomposition methods for multistage stochastic programming problems

We consider well-known decomposition techniques for multistage stochastic programming and a new scheme based on normal solutions for stabilizing iterates during the solution process. The given algorithms combine ideas from finite perturbation of convex programs and level bundle methods to regularize the so-called forward step of these decomposition methods. Numerical experiments on a hydrothermal scheduling problem indicate that our algorithms are competitive with the state-of-the-art approaches such as multistage regularized decomposition, nested decomposition and stochastic dual dynamic programming.

非光滑约束优化基于非精确数据的改进水平束方法

非光滑约束优化基于非精确数据的改进水平束方法

Target radius methods for nonsmooth convex optimization

We consider the target radius method of A. Ouorou (2009) and propose some extensions to handle particular classes of (a) nonsmooth convex (nonlinearly) constrained optimization problems; (b) bilevel optimization problems. The given approaches generalize the well-known level bundle methods, and one of the new algorithms possesses (nearly) dimension-independent iteration complexity.

BUNDLE METHODS IN THE XXIst CENTURY: A BIRD'S-EYE VIEW

Bundle methods are often the algorithms of choice for nonsmooth convex optimization, especially if accuracy in the solution and reliability are a concern. We review several algorithms based on the bundle methodology that have been developed recently and that, unlike their forerunner variants, have the ability to provide exact solutions even if most of the time the available information is inaccurate. We adopt an approach that is by no means exhaustive, but covers different proximal and level bundle methods dealing with inexact oracles, for both unconstrained and constrained problems.

Level bundle methods for oracles with on-demand accuracy

For non-smooth convex optimization, we consider level bundle methods built using an oracle that computes values for the objective function and a subgradient at any given feasible point. For the problems of interest, the exact oracle information is computable, but difficult to obtain. In order to save computational effort the oracle can provide estimations with an accuracy that depends on two additional parameters, informed to the oracle together with the evaluation point. The first of such parameters is a descent target, while the second one is a bound for inexactness. If the oracle can reach the target with its function estimation, then the corresponding error is bounded by the second parameter. Otherwise, if the oracle detects that the target cannot be met, the function and subgradient estimations can be rough and have an unknown accuracy. For some selected iterates the considered methods drive the inexactness parameter to zero, thus ensuring that an exact solution to the optimization problem is asymptotically found. The approach is comprehensive and covers known exact and inexact level methods as well as some novel variants that can handle inaccuracy in an adaptive manner. In particular, when the feasible set is also compact, some of the new on-demand accuracy methods have the same rate of convergence of exact level variants known in the literature. A numerical benchmark on a battery of two-stage stochastic linear programs assesses the interest of the approach, substantially faster than the L-shaped method, without any accuracy loss.

An Approximate Sequential Bundle Method for Solving a Convex Nondifferentiable Bilevel Programming Problem

By combining two bundle methods, PBMASL (proximal bundle method with approximate subgradient linearizations) and DPLBM (descent proximal level bundle method), we present an approximate sequential bundle algorithm for solving a bilevel programming problem with a nondifferentiable convex objective function and two separable constraints. In the proposed algorithm, the values of the objective function in the constraints and its subgradients are computed approximately, the estimates of the tolerances are not required for convergence proof. The presented results improve and extend the earlier work.

A Cutting Plane and Level Stabilization Bundle Method with Inexact Data for Minimizing Nonsmooth Nonconvex Functions

Under the condition that the values of the objective function and its subgradient are computed approximately, we introduce a cutting plane and level bundle method for minimizing nonsmooth nonconvex functions by combining cutting plane method with the ideas of proximity control and level constraint. The proposed algorithm is based on the construction of both a lower and an upper polyhedral approximation model to the objective function and calculates new iteration points by solving a subproblem in which the model is employed not only in the objective function but also in the constraints. Compared with other proximal bundle methods, the new variant updates the lower bound of the optimal value, providing an additional useful stopping test based on the optimality gap. Another merit is that our algorithm makes a distinction between affine pieces that exhibit a convex or a concave behavior relative to the current iterate. Convergence to some kind of stationarity point is proved under some looser conditions.

Level bundle methods for constrained convex optimization with various oracles

We propose restricted memory level bundle methods for minimizing constrained convex nonsmooth optimization problems whose objective and constraint functions are known through oracles (black-boxes) that might provide inexact information. Our approach is general and covers many instances of inexact oracles, such as upper, lower and on-demand accuracy oracles. We show that the proposed level bundle methods are convergent as long as the memory is restricted to at least four well chosen linearizations: two linearizations for the objective function, and two linearizations for the constraints. The proposed methods are particularly suitable for both joint chance-constrained problems and two-stage stochastic programs with risk measure constraints. The approach is assessed on realistic joint constrained energy problems, arising when dealing with robust cascaded-reservoir management.

Proximal level bundle methods for convex nondifferentiable optimization, saddle-point problems and variational inequalities

We study proximal level methods for convex optimization that use projections onto successive approximations of level sets of the objective corresponding to estimates of the optimal value. We show that they enjoy almost optimal efficiency estimates. We give extensions for solving convex constrained problems, convex-concave saddle-point problems and variational inequalities with monotone operators. We present several variants, establish their efficiency estimates, and discuss possible implementations. In particular, our methods require bounded storage in contrast to the original level methods of Lemarechal, Nemirovskii and Nesterov.

A descent proximal level bundle method for convex nondifferentiable optimization

We give a proximal level method for convex minimization that uses projections onto successive approximations of level sets of the objective. In contrast to the original level methods of Lemarechal, Nemirovskii and Nesterov, our method is globally convergent without any compactness assumptions and requires bounded storage. It does not employ potentially expensive linesearches as does the level method of Brännlund.