Ann-Brith Strömberg

Ergodic, primal convergence in dual subgradient schemes for convex programming

Abstract.Lagrangean dualization and subgradient optimization techniques are frequently used within the field of computational optimization for finding approximate solutions to large, structured optimization problems. The dual subgradient scheme does not automatically produce primal feasible solutions; there is an abundance of techniques for computing such solutions (via penalty functions, tangential approximation schemes, or the solution of auxiliary primal programs), all of which require a fair amount of computational effort.We consider a subgradient optimization scheme applied to a Lagrangean dual formulation of a convex program, and construct, at minor cost, an ergodic sequence of subproblem solutions which converges to the primal solution set. Numerical experiments performed on a traffic equilibrium assignment problem under road pricing show that the computation of the ergodic sequence results in a considerable improvement in the quality of the primal solutions obtained, compared to those generated in the basic subgradient scheme.

Conditional subgradient optimization — Theory and applications

Abstract We generalize the subgradient optimization method for nondifferentiable convex programming to utilize conditional subgradients. Firstly, we derive the new method and establish its convergence by generalizing convergence results for traditional subgradient optimization. Secondly, we consider a particular choice of conditional subgradients, obtained by projections, which leads to an easily implementable modification of traditional subgradient optimization schemes. To evaluate the subgradient projection method we consider its use in three applications: uncapacitated facility location, two-person zero-sum matrix games, and multicommodity network flows. Computational experiments show that the subgradient projection method performs better than traditional subgradient optimization; in some cases the difference is considerable. These results suggest that our simply modification may improve subgradient optimization schemes significantly. This finding is important as such schemes are very popular, especially in the context of Lagrangean relaxation.

On the convergence of conditional ε-subgradient methods for convex programs and convex–concave saddle-point problems

Abstract The paper provides two contributions. First, we present new convergence results for conditional e -subgradient algorithms for general convex programs. The results obtained here extend the classical ones by Polyak [Sov. Math. Doklady 8 (1967) 593; USSR Comput. Math. Math. Phys. 9 (1969) 14; Introduction to Optimization, Optimization Software, New York, 1987] as well as the recent ones in [Math. Program. 62 (1993) 261; Eur. J. Oper. Res. 88 (1996) 382; Math. Program. 81 (1998) 23] to a broader framework. Secondly, we establish the application of this technique to solve non-strictly convex–concave saddle point problems, such as primal-dual formulations of linear programs. Contrary to several previous solution algorithms for such problems, a saddle-point is generated by a very simple scheme in which one component is constructed by means of a conditional e -subgradient algorithm, while the other is constructed by means of a weighted average of the (inexact) subproblem solutions generated within the subgradient method. The convergence result extends those of [Minimization Methods for Non-Differentiable Functions, Springer-Verlag, Berlin, 1985; Oper. Res. Lett. 19 (1996) 105; Math. Program. 86 (1999) 283] for Lagrangian saddle-point problems in linear and convex programming, and of [Int. J. Numer. Meth. Eng. 40 (1997) 1295] for a linear–quadratic saddle-point problem arising in topology optimization in contact mechanics.

Ergodic convergence in subgradient optimization

When nonsmooth, convex minimization problems are solved by subgradient optimization methods, the subgradients used will in general not accumulate to subgradients which verify the optimality of a solution obtained in the limit. It is therefore not a straightforward task to monitor the progress of a subgradient method in terms of the approximate fulfillment of optimality conditions. Further, certain supplementary information, such as convergent estimates of Lagrange multipliers and convergent lower bounds on the optimal objective value, is not directly available in subgradient schemes As a means of overcoming these weaknesses in subgradient methods, we introduce the computation of an ergodic (averaged) sequence of subgradients. Specifically, we consider a nonsmooth, convex program solved by a conditional subgradient optimization scheme with divergent series step lengths, and show that the elements of the ergodic sequence of subgradients in the limit fulfill the optimality conditions at the optimal solution,...

Benefits of using an optimization methodology for identifying robust process integration investments under uncertainty--A pulp mill example

This paper presents a case study on the optimization of process integration investments in a pulp mill considering uncertainties in future electricity and biofuel prices and CO2 emissions charges. The work follows the methodology described in Svensson et al. [Svensson, E., Berntsson, T., Stromberg, A.-B., Patriksson, M., 2008b. An optimization methodology for identifying robust process integration investments under uncertainty. Energy Policy, in press, doi:10.1016/j.enpol.2008.10.023] where a scenario-based approach is proposed for the modelling of uncertainties. The results show that the proposed methodology provides a way to handle the time dependence and the uncertainties of the parameters. For the analyzed case, a robust solution is found which turns out to be a combination of two opposing investment strategies. The difference between short-term and strategic views for the investment decision is analyzed and it is found that uncertainties are increasingly important to account for as a more strategic view is employed. Furthermore, the results imply that the obvious effect of policy instruments aimed at decreasing CO2 emissions is, in applications like this, an increased profitability for all energy efficiency investments, and not as much a shift between different alternatives.

Primal convergence from dual subgradient methods for convex optimization

When solving a convex optimization problem through a Lagrangian dual reformulation subgradient optimization methods are favorably utilized, since they often find near-optimal dual solutions quickly. However, an optimal primal solution is generally not obtained directly through such a subgradient approach unless the Lagrangian dual function is differentiable at an optimal solution. We construct a sequence of convex combinations of primal subproblem solutions, a so called ergodic sequence, which is shown to converge to an optimal primal solution when the convexity weights are appropriately chosen. We generalize previous convergence results from linear to convex optimization and present a new set of rules for constructing the convexity weights that define the ergodic sequence of primal solutions. In contrast to previously proposed rules, they exploit more information from later subproblem solutions than from earlier ones. We evaluate the proposed rules on a set of nonlinear multicommodity flow problems and demonstrate that they clearly outperform the ones previously proposed.

Approximating the Pareto optimal set using a reduced set of objective functions

Real-world applications of multi-objective optimization often involve numerous objective functions. But while such problems are in general computationally intractable, it is seldom necessary to determine the Pareto optimal set exactly. A significantly smaller computational burden thus motivates the loss of precision if the size of the loss can be estimated. We describe a method for finding an optimal reduction of the set of objectives yielding a smaller problem whose Pareto optimal set w.r.t. a discrete subset of the decision space is as close as possible to that of the original set of objectives. Utilizing a new characterization of Pareto optimality and presuming a finite decision space, we derive a program whose solution represents an optimal reduction. We also propose an approximate, computationally less demanding formulation which utilizes correlations between the objectives and separates into two parts. Numerical results from an industrial instance concerning the configuration of heavy-duty trucks are also reported, demonstrating the usefulness of the method developed. The results show that multi-objective optimization problems can be significantly simplified with an induced error which can be measured.

The stochastic opportunistic replacement problem, part II: a two-stage solution approach

In Almgren et al. (The opportunistic replacement problem: analysis and case studies, preprint, Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, Göteborg, Sweden, 2011) we studied the opportunistic replacement problem, which is a multi-component maintenance scheduling problem with deterministic component lives. The assumption of deterministic lives is a substantial simplification, but valid in applications where critical components are assigned a technical life after which replacement is enforced. Here, we study the stochastic opportunistic replacement problem, which is a more general setting in which component lives are allowed to be stochastic. We consider a stochastic programming approach for the minimization of the expected cost over the remaining planning horizon. Further, we present a means to compute lower bounds on the recourse function. The lower bounds are used in the construction of a decomposition method which extends the integer L-shaped decomposition method to incorporate stronger optimality cuts. In order to obtain a computationally tractable model, a two-stage sample average approximation scheme is utilized. Numerical experiments on problem instances from the wind power and aviation industry as well as on two test instances are performed. The results show that the decomposition method is faster than solving the deterministic equivalent on all four instances considered. Furthermore, the numerical experiments show that decisions based on the stochastic programming approach compared with simpler maintenance policies yield maintenance decisions with a significantly lower expected total maintenance cost on two out of the four instances tested, and an equivalent maintenance cost compared to the best policy on the remaining two instances.

A stochastic model for opportunistic maintenance planning of offshore wind farms

A sound maintenance planning is of crucial importance for wind power farms, and especially for offshore locations. This paper presents a stochastic optimization model for opportunistic service maintenance of offshore wind farms. The model takes advantage of 7 days wind production ensemble forecast and opportunities at corrective maintenance activities in order to perform the service maintenance tasks at the lowest cost. The model is based on a rolling horizon, i.e. the optimization is performed on a daily basis to update the maintenance planning based on the updated production and weather forecasts. An example based on real wind data is used to demonstrate the value of the proposed approach. In this example, it is shown that 32% of the cost for production losses and transportation could be saved.

An Optimal Number-Dependent Preventive Maintenance Strategy for Offshore Wind Turbine Blades Considering Logistics

In offshore wind turbines, the blades are among the most critical and expensive components that suffer from different types of damage due to the harsh maritime environment and high load. The blade damages can be categorized into two types: the minor damage, which only causes a loss in wind capture without resulting in any turbine stoppage, and the major (catastrophic) damage, which stops the wind turbine and can only be corrected by replacement. In this paper, we propose an optimal number-dependent preventive maintenance (NDPM) strategy, in which a maintenance team is transported with an ordinary or expedited lead time to the offshore platform at the occurrence of the th minor damage or the first major damage, whichever comes first. The long-run expected cost of the maintenance strategy is derived, and the necessary conditions for an optimal solution are obtained. Finally, the proposed model is tested on real data collected from an offshore wind farm database. Also, a sensitivity analysis is conducted in order to evaluate the effect of changes in the model parameters on the optimal solution.