| 1. |
- Almgren, Torgny, 1962, et al.
(författare)
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Optimization models for improving periodic maintenance schedules by utilizing opportunities
- 2012
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Ingår i: Proceedings of 4th Production and Operations Management World Conference, July 2012.
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Konferensbidrag (refereegranskat)abstract
- We present mathematical models for finding optimal opportunistic maintenance schedules for systems, in which components are assigned maximum replacement intervals. Our mod- els are applied to safety-critical components in an aircraft engine, for which maintenance opportunities naturally arise since entire modules are sent to the workshop when mainte- nance is required on one or more components. Case study results illustrate the advantage of the mathematical models over simpler policies, the benefit of coordinating the maintenance in economically dependent systems, and that our models can be utilized also for strategic investment decision support.
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| 2. |
- Almgren, Torgny, 1962, et al.
(författare)
-
Optimization of opportunistic replacement activities: A case study in the aircraft industry
- 2007
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- In the aircraft industry maximizing availability is essential. Maintenance schedules must therefore be opportunistic, incorporating preventive maintenance activities within the scheduled as well as the unplanned ones. At the same time, the maintenance contractor should utilize opportunistic maintenance to enable the minimization of the total expected cost to have a functional aircraft engine and thus to provide attractive service contracts. This paper provides an opportunistic maintenance optimization model which has been constructed and tested together with Volvo Aero Corporation in Trollhättan, Sweden for the maintenance of the RM12 engine. The model incorporates components with deterministic as well as with stochastic lives. The replacement model is shown to have favourable properties; in particular, when the maintenance occasions are fixed the remaining problem has the integrality property, the replacement polytope corresponding to the convex hull of feasible solutions is full-dimensional, and all the necessary constraints for its definition are facet-inducing. We present an empirical crack growth model that estimates the remaining life and also a case study that indicates that a non-stationary renewal process with Weibull distributed lives is a good model for the recurring maintenance occasions. Using one point of support for the distribution yields a deterministic replacement model; it is evaluated against classic maintenance policies from the literature through stochastic simulations. The deterministic model provides maintenance schedules over a finite time period that induce fewer maintenance occasions as well as fewer components replaced.
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| 3. |
- Almgren, Torgny, 1962, et al.
(författare)
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The opportunistic replacement problem: analysis and case studies
- 2011
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We consider an optimization model for determining optimal opportunistic maintenance (that is, component replacement) schedules when data is deterministic. This problem generalizes that of Dickman, Epstein, and Wilamowsky [21] and is a natural starting point for the modelling of replacement schedules when component lives are non-deterministic. We show that this basic opportunistic replacement problem is NP-hard. We show that the convex hull of the set of feasible replacement schedules is full-dimensional, and that all the necessary inequalities also are facet-inducing. We show that when maintenance occasions are fixed, the remaining problem can be stated as a linear program; when maintenance costs are monotone with time, the latter is solvable through a greedy procedure. Results from a series of case studies performed in the areas of aircraft engine and wind turbine maintenance are also reported. These illustrate the advantages of utilizing opportunistic maintenance activities based on a complete optimization model, as compared to simpler policies.
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| 4. |
- Almgren, Torgny, 1962, et al.
(författare)
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The opportunistic replacement problem: theoretical analyses and numerical tests
- 2012
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Ingår i: Mathematical Methods of Operations Research. - : Springer Science and Business Media LLC. - 1432-2994 .- 1432-5217. ; 76:3, s. 289-319
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Tidskriftsartikel (refereegranskat)abstract
- We consider a model for determining optimal opportunistic maintenance schedules with respect to a maximum replacement interval. This problem generalizes that of Dickman et al. (J Oper Res Soc India 28:165–175, 1991) and is a natural starting point for modelling replacement schedules of more complex systems. We show that this basic opportunistic replacement problem is NP-hard, that the convex hull of the set of feasible replacement schedules is full-dimensional, that all the inequalities of the model are facet-inducing, and present a new class of facets obtained through a {0,1/2}-Chvátal–Gomory rounding. For costs monotone with time, a class of elimination constraints is introduced to reduce the computation time; it allows maintenance only when the replacement of at least one component is necessary. For costs decreasing with time, these constraints eliminate non-optimal solutions. When maintenance occasions are fixed, the remaining problem is stated as a linear program and solved by a greedy procedure. Results from a case study on aircraft engine maintenance illustrate the advantage of the optimization model over simpler policies. We include the new class of facets in a branch-and-cut framework and note a decrease in the number of branch-and-bound nodes and simplex iterations for most instance classes with time dependent costs. For instance classes with time independent costs and few components the elimination constraints are used favorably. For fixed maintenance occasions the greedy procedure reduces the computation time as compared with linear programming techniques for all instances tested.
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| 5. |
- Almgren, Torgny, 1962, et al.
(författare)
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The replacement problem: A polyhedral and complexity analysis. The complete version
- 2009
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We consider an optimization model for determining optimal opportunistic maintenance (that is, component replacement) schedules when data is deterministic. This problem, which generalizes that of Dickman et al., is a natural starting point for the modelling of replacement schedules when component lives are non-deterministic, whence a mathematical study of the model is of large interest. We show that the convex hull of the set of feasible replacement schedules is full-dimensional, and that all the necessary inequalities are facet-inducing. Additional facets are then provided through Chvatal-Gomory rounding. We show that when maintenance occasions are fixed, the remaining problem reduces to a linear program; in some cases the latter is solvable through a greedy procedure. We further show that this basic replacement problem is NP-hard.
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| 6. |
- Besnard, Francois, et al.
(författare)
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An Optimization Framework for Opportunistic Maintenance of Offshore Wind Power System
- 2009
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Ingår i: 2009 IEEE BUCHAREST POWERTECH, VOLS 1-5. - NEW YORK : IEEE. - 9781424422340 ; , s. 2970-2976
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Konferensbidrag (refereegranskat)abstract
- A sound maintenance planning is of crucial importance for wind power farms, and especially for offshore locations. There is a large potential in cost savings by maintenance optimization to make the projects more cost-efficient. This paper presents an opportunistic maintenance optimization model for offshore wind power system. The model takes advantage of wind forecasts and corrective maintenance activities in order to perform preventive maintenance tasks at low costs. The approach is illustrated with an example to demonstrate the value of the optimization. In this example 43% of the cost to perform preventive maintenance could be saved using the proposed method.
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| 7. |
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| 8. |
- Eriksson Barman, Sandra, 1985, et al.
(författare)
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Modeling and solving vehicle routing problems with many available vehicle types
- 2015
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Ingår i: Springer Proceedings in Mathematics & Statistics. - Cham : Springer International Publishing. - 2194-1009 .- 2194-1017. - 9783319185668
-
Konferensbidrag (refereegranskat)abstract
- Vehicle routing problems (VRP) involving the selection of vehicles from a large set of vehicle types are hitherto not well-studied in the literature. Such problems arise at Volvo Group Trucks Technology, who faces an immense set of possible vehicle configurations, of which an optimal set needs to be chosen for each specific combination of transport missions. Another property of real-world VRP’s that is often neglected in the literature is that the fuel resources required to drive a vehicle along a route is highly dependent on the actual load of the vehicle. We define the fleet size and mix VRP with many available vehicle types, called many-FSMVRP, and suggest an extended set-partitioning model of this computationally demanding combinatorial optimization problem. To solve the extended model, we have developed a method based on Benders’ decomposition, the subproblems of which are solved using column generation, and the column generation subproblems being solved using dynamic programming; the method is implemented with a so-called projection-of-routes procedure. The resulting method is compared with a column generation approach for the standard set-partitioning model. Our method for the extended model performs on par with column generation applied to the standard model for instances such that the two models are equivalent. In addition, the utility of the extended model for instances with very many available vehicle types is demonstrated. Our method is also shown to efficiently handle cases in which the costs are dependent on the load of the vehicle. Computational tests on a set of extended standard test instances show that our method, based on Benders’ algorithm, is able to determine combinations of vehicles and routes that are optimal to a relaxation (w.r.t. the route decision variables) of the extended model. Our exact implementation of Benders’ algorithm appears, however, too slow when the number of customers grows. To improve its performance, we suggest that relaxed versions of the column generation subproblems are solved, and that the set-partitioning model is replaced by a set-covering model.
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| 9. |
- Fotedar, Sunney, 1989, et al.
(författare)
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A criterion space decomposition approach to generalized tri-objective tactical resource allocation
- 2023
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Ingår i: Computational Management Science. - : Springer Science and Business Media LLC. - 1619-697X .- 1619-6988. ; 20:1
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Tidskriftsartikel (refereegranskat)abstract
- We present a tri-objective mixed-integer linear programming model of the tactical resource allocation problem with inventories, called the generalized tactical resource allocation problem (GTRAP). We propose a specialized criterion space decomposition strategy, in which the projected two-dimensional criterion space is partitioned and the corresponding sub-problems are solved in parallel by application of the quadrant shrinking method (QSM) (Boland in Eur J Oper Res 260(3):873–885, 2017) for identifying non-dominated points. To obtain an efficient implementation of the parallel variant of the QSM we suggest some modifications to reduce redundancies. Our approach is tailored for the GTRAP and is shown to have superior computational performance as compared to using the QSM without parallelization when applied to industrial instances.
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| 10. |
- Fotedar, Sunney, et al.
(författare)
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Bi-objective optimization of the tactical allocation of job types to machines: mathematical modeling, theoretical analysis, and numerical tests
- 2022
-
Ingår i: International Transactions in Operational Research. - : Wiley. - 0969-6016 .- 1475-3995.
-
Tidskriftsartikel (refereegranskat)abstract
- We introduce a tactical resource allocation model for a large aerospace engine system manufacturer aimed at long-term production planning. Our model identifies the routings a product takes through the factory, and which machines should be qualified for a balanced resource loading, to reduce product lead times. We prove some important mathematical properties of the model that are used to develop a heuristic providing a good initial feasible solution. We propose a tailored approach for our class of problems combining two well-known criterion space search algorithms, the bi-directional epsilon-constraint method and the augmented weighted Tchebycheff method. A computational investigation comparing solution times for several solution methods is presented for 60 numerical instances.
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| 11. |
- Fotedar, Sunney, et al.
(författare)
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Robust optimization of a bi-objective tactical resource allocation problem with uncertain qualification costs
- 2022
-
Ingår i: Autonomous Agents and Multi-Agent Systems. - : Springer Science and Business Media LLC. - 1387-2532 .- 1573-7454. ; 36:2
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Tidskriftsartikel (refereegranskat)abstract
- In the presence of uncertainties in the parameters of a mathematical model, optimal solutions using nominal or expected parameter values can be misleading. In practice, robust solutions to an optimization problem are desired. Although robustness is a key research topic within single-objective optimization, little attention is received within multi-objective optimization, i.e. robust multi-objective optimization.This work builds on recent work within robust multi-objective optimization and presents a new robust efficiency concept for bi-objective optimization problems with one uncertain objective. Our proposed concept and algorithmic contribution are tested on a real-world multi-item capacitated resource planning problem, appearing at a large aerospace company manufacturing high precision engine parts. Our algorithm finds all the robust efficient solutions required by the decision-makers in significantly less time than the approach of Kuhn et al. (Eur J Oper Res 252(2):418-431, 2016) on 28 of the 30 industrial instances.
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| 12. |
- Granfeldt, Caroline, 1991, et al.
(författare)
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A Lagrangian relaxation approach to an electricity system investment model with a high temporal resolution
- 2023
-
Ingår i: OR Spectrum. - 1436-6304 .- 0171-6468. ; 45:4, s. 1263-1294
-
Tidskriftsartikel (refereegranskat)abstract
- The global production of electricity contributes significantly to the release of carbon dioxide emissions. Therefore, a transformation of the electricity system is of vital importance in order to restrict global warming. This paper proposes a modelling methodology for electricity systems with a large share of variable renewable electricity generation, such as wind and solar power. The model developed addresses the capacity expansion problem, i.e. identifying optimal long-term investments in the electricity system. Optimal investments are defined by minimum investment and production costs under electricity production constraints—having different spatial resolutions and technical detail—while meeting the electricity demand. Our model is able to capture a range of strategies to manage variations and to facilitate the integration of variable renewable electricity; it is very large due to the high temporal resolution required to capture the variations in wind and solar power production and the chronological time representation needed to model energy storage. Moreover, the model can be further extended—making it even larger—to capture a large geographical scope, accounting for the trade of electricity between regions with different conditions for wind and solar power. Models of this nature thus typically need to be solved using some decomposition method to reduce solution times. In this paper, we develop a decomposition method using so-called variable splitting and Lagrangian relaxation; the dual problem is solved by a deflected subgradient algorithm. Our decomposition regards the temporal resolution by defining 2-week periods throughout the year and relaxing the overlapping constraints. The method is tested and evaluated on some real-world cases containing regions with different energy mixes and conditions for wind power. Numerical results show shorter computation times as compared with the non-decomposed model, and capacity investment options similar to the optimal solution provided by the latter model.
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| 13. |
- Gustavsson, Emil, 1987, et al.
(författare)
-
Preventive maintenance scheduling of multi-component systems with deterioration costs
- 2012
-
Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We introduce the preventive maintenance scheduling problem with interval costs (PMSPIC), which is to schedule preventive maintenance of components of a system over a finite discretized time horizon, given a common set-up cost and component costs dependent on the lengths of the maintenance intervals. We present a 0-1 integer linear programming (0-1 ILP) model for the PMSPIC which was originally presented by Joneja (1990) to model the joint replenishment problem. We show that most of the integrality constraints can be relaxed and that the linear inequality constraints define facets of the convex hull of the feasible set. We present three applications demonstrating that the PMSPIC can be used to model several types of maintenance problems with deterioration costs. The first considers rail grinding. If the interval between the grinding occasions increases, then the sizes of the rail cracks increase, which implies that more grinding passes must be performed, generating a higher maintenance cost. We presume a deterministic model for crack growth and optimize the scheduling of the rail grinding on a set of track sections. Our second application concerns two approaches for the scheduling of component replacements in aircraft engines. In the first approach a bi-objective problem, simultaneously minimizing the cost for the scheduled preventive maintenance and the probability of unexpected stops, is formulated. The second approach considers the minimization of the sum of costs of preventive and expected corrective maintenance, without rescheduling. We also demonstrate that if rescheduling is allowed, then the 0-1 ILP model can be used as a policy by re-optimizing the schedule at a component failure, thus utilizing the opportunity for preventive maintenance. We evaluate the use of such a strategy in a simulation of the engine. The third approach considers components’ replacement in wind mills in a wind farm, extending the PMSPIC to consider several systems with a joint set-up cost. As for the aircraft engine application, we use the 0-1 ILP model as a policy for deciding upon replacement decisions allowing for rescheduling, and evaluate it by simulating the joint system. In each of the three applications, the use of the 0-1 ILP model is compared with age or constantinterval policies, resulting in a reduction of maintenance costs by up to 15% compared with the respective best simple policy.
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| 14. |
- Gustavsson, Emil, 1987, et al.
(författare)
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Preventive maintenance scheduling of multi-component systems with interval costs
- 2014
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Ingår i: Computers & industrial engineering. - : Elsevier BV. - 0360-8352. ; 76, s. 390-400
-
Tidskriftsartikel (refereegranskat)abstract
- We introduce the preventive maintenance scheduling problem with interval costs (PMSPIC), which is to schedule preventive maintenance (PM) of the components of a system over a finite and discretized time horizon, given a common set-up cost and component costs dependent on the lengths of the maintenance intervals. We present a 0–1 integer linear programming (0–1 ILP) model for the PMSPIC; the model is identical to that presented by Joneja (1990) for the joint replenishment problem within inventory management. We study this model from a polyhedral and exact solutions’ point of view, as opposed to previously studied heuristics (e.g. Boctor, Laporte, & Renaud, 2004; Federgruen & Tzur, 1994; Levi, Roundy, & Shmoys, 2006; Joneja, 1990).We show that most of the integrality constraints can be relaxed and that the linear inequality constraints define facets of the convex hull of the feasible set. We further relate the PMSPIC to the opportunistic replacement problem, for which detailed polyhedral studies were performed by Almgren et al. (2012a). The PMSPIC can be used as a building block to model several types of maintenance planning problems possessing deterioration costs. By a careful modeling of these costs, a polyhedrally sound 0–1 ILP model is used to find optimal solutions to realistic-sized multi-component maintenance planning problems. The PMSPIC is thus easily extended by side constraints or to multiple tiers, which is demonstrated through three applications; these are chosen to span several levels of unmodeled randomness requiring fundamentally different maintenance policies, which are all handled by variations of our basic model. Our first application considers rail grinding. Rail cracks increase with increasing intervals between grinding occasions, implying that more grinding passes must be performed—thus generating higher costs. We optimize the grinding schedule for a set of track sections presuming a deterministic model for crack growth; hence, no corrective maintenance (CM) will occur between the grinding occasions scheduled. The second application concerns two approaches for scheduling component replacements in aircraft engines. The first approach is bi-objective, simultaneously minimizing the cost for the scheduled PM and the probability of unexpected stops. In the second approach the sum of costs for PM and expected CM—without rescheduling—is minimized. When rescheduling is allowed, the 0–1 ILP model is used as a policy by re-optimizing the schedule at a component failure, which then constitutes an opportunity for PM. The policy manages the trade-off between costs for PM and unplanned CM and is evaluated in a simulation of the engine. The third application considers components’ replacement in wind mills in a wind farm, extending the PMSPIC to comprise multiple tiers with joint set-up costs. Due to the large number of components unexpected stops occur frequently, thus calling for a dynamic rescheduling, which is evaluated through a simulation of the system. In each of the three applications, the use of the 0-1 ILP model is compared with age or constant-interval policies; the maintenance costs are reduced by up to 16% as compared with the respective best simple policy. The results are strongest for the first two applications, possessing low levels of unmodeled randomness.
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| 15. |
- Gustavsson, Emil, 1987, et al.
(författare)
-
Primal convergence from dual subgradient methods for convex optimization
- 2012
-
Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- When solving a convex optimization problem through a Lagrangian dual reformulation subgradient optimization methods are favourably 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. 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.
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| 16. |
- Gustavsson, Emil, 1987, et al.
(författare)
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Primal convergence from dual subgradient methods for convex optimization
- 2015
-
Ingår i: Mathematical Programming. - : Springer Science and Business Media LLC. - 0025-5610 .- 1436-4646. ; 150:2, s. 365-390
-
Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 17. |
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| 18. |
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| 19. |
- Laksman, Efraim, 1983, et al.
(författare)
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The stochastic opportunistic replacement problem, part III: improved bounding procedures
- 2020
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Ingår i: Annals of Operations Research. - : Springer Science and Business Media LLC. - 1572-9338 .- 0254-5330. ; 292:2, s. 711-733
-
Tidskriftsartikel (refereegranskat)abstract
- We consider the problem to find a schedule for component replacement in a multi-component system, whose components possess stochastic lives and economic dependencies, such that the expected costs for maintenance during a pre-defined time period are minimized. The problem was considered in Patriksson et al. (Ann Oper Res 224:51–75, 2015), in which a two-stage approximation of the problem was optimized through decomposition (denoted the optimization policy). The current paper improves the effectiveness of the decomposition approach by establishing a tighter bound on the value of the recourse function (i.e., the second stage in the approximation). A general lower bound on the expected maintenance cost is also established. Numerical experiments with 100 simulation scenarios for each of four test instances show that the tighter bound yields a decomposition generating fewer optimality cuts. They also illustrate the quality of the lower bound. Contrary to results presented earlier, an age-based policy performs on par with the optimization policy, although most simple policies perform worse than the optimization policy.
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| 20. |
- Larsson, Torbjörn, et al.
(författare)
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Ergodic convergence in subgradient optimization - with application to simplicial decomposition of convex programs
- 2012
-
Ingår i: Contemporary Mathematics. - Providence, Rhode Island : American Mathematical Society. - 0271-4132 .- 1098-3627. ; 568, s. 159-186
-
Tidskriftsartikel (refereegranskat)abstract
- When non-smooth, convex minimization problems are solved by subgradient optimization methods, the subgradients used will in general not accumulate to subgradients that verify the optimality of a solution obtained in the limit. It is therefore not a straightforward task to monitor the progress of subgradient methods in terms of the approximate fulfilment 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 introduced in LPS96b, LPS96c, and LPS98 the computation of an ergodic (averaged) sequence of subgradients. Specifically, we considered a non-smooth, convex program solved by a conditional subgradient optimization scheme with divergent series step lengths, and showed that the elements of the ergodic sequence of subgradients in the limit fulfil the optimality conditions at the optimal solution, to which the sequence of iterates converges. This result has three important implications. The first is the finite identification of active constraints at the solution obtained in the limit. The second is the establishment of the convergence of ergodic sequences of Lagrange multipliers; this result enables sensitivity analyses for solutions obtained by subgradient methods. The third is the convergence of a lower bounding procedure based on an ergodic sequence of affine underestimates of the objective function; this procedure also provides a proper termination criterion for subgradient optimization methods. This article contributes first an overview of results and applications found in LPS96b, LPS96c, and LPS98 pertaining to the generation of ergodic sequences of subgradients generated within a subgradient scheme. It then presents an application of these results to that of the first instance of a simplicial decomposition algorithm for convex and non-smooth optimization problems.
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| 21. |
- Larsson, Torbjörn, et al.
(författare)
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Ergodic results and bounds on the optimal value in subgradient optimization
- 1996
-
Ingår i: Operations Research Proceedings 1995. - 9783540608066 ; , s. 30-35
-
Konferensbidrag (refereegranskat)abstract
- Subgradient methods are popular tools for nonsmooth, convex minimization, especially in the context of Lagrangean relaxation; their simplicity has been a main contribution to their success. As a consequence of the nonsmoothness, it is not straightforward to monitor the progress of a subgradient method in terms of the approximate fulfilment of optimality conditions, since the subgradients used in the method will, in general, not accumulate to subgradients that verify optimality of a solution obtained in the limit. Further, certain supplementary information, such as convergent estimates of Lagrange multipliers, is not directly available in subgradient schemes. As a means for overcoming these weaknesses of subgradient optimization 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 (of which the traditional sub gradient optimization method is a special case) with divergent series step lengths, which generates a sequence of iterates that converges to an optimal solution. We show that the elements of the ergodic sequence of subgradients in the limit fulfill the optimality conditions at this optimal solution. Further, we use the convergence properties of the ergodic sequence of subgradients to establish convergence of an ergodic sequence of Lagrange multipliers. Finally, some potential applications of these ergodic results are briefly discussed.
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| 22. |
- Larsson, Torbjörn, et al.
(författare)
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Ergodic results in subgradient optimization
- 1996
-
Ingår i: Nonlinear Optimization and Applications. - 9780306453168 ; , s. 229-248
-
Konferensbidrag (refereegranskat)abstract
- Subgradient methods are popular tools for nonsmooth, convex minimization, especially in the context of Lagrangean relaxation; their simplicity has been a main contribution to their success. As a consequence of the nonsmoothness, it is not straightforward to monitor the progress of a subgradient method in terms of the approximate fulfilment of optimality conditions, since the subgradients used in the method will, in general, not accumulate to subgradients that verify optimality of a solution obtained in the limit. Further, certain supplementary information, such as convergent estimates of Lagrange multipliers, is not directly available in subgradient schemes. As a means for overcoming these weaknesses of subgradient optimization 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 (of which the traditional subgradient optimization method is a special case) with divergent series step lengths, which generates a sequence of iterates that converges to an optimal solution. We show that the elements of the ergodic sequence of subgradients in the limit fulfill the optimality conditions at this optimal solution. Further, we use the convergence properties of the ergodic sequence of subgradients to establish convergence of an ergodic sequence of Lagrange multipliers. Finally, some potential applications of these ergodic results are briefly discussed.
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| 23. |
- Lindroth, Peter, 1979, et al.
(författare)
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Approximating the Pareto Optimal Set using a Reduced Set of Objective Functions
- 2010
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Ingår i: European Journal of Operational Research. - : Elsevier BV. - 0377-2217. ; 207:3, s. 1519-1534
-
Tidskriftsartikel (refereegranskat)abstract
- 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.
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| 24. |
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| 25. |
- Nedelkova, Zuzana, 1987, et al.
(författare)
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A splitting algorithm for simulation-based optimization problems with categorical variables
- 2019
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Ingår i: Engineering Optimization. - 1029-0273 .- 0305-215X. ; 51:5, s. 815-831
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Tidskriftsartikel (refereegranskat)abstract
- In the design of complex products, some product components can only be chosen from a finite set of options. Each option then corresponds to a multidimensional point representing the specifications of the chosen components. A splitting algorithm that explores the resulting discrete search space and is suitable for optimization problems with simulation-based objective functions is presented. The splitting rule is based on the representation of a convex relaxation of the search space in terms of a minimum spanning tree and adopts ideas from multilevel coordinate search. The objective function is underestimated on its domain by a convex quadratic function. The main motivation is the aim to find—for a vehicle and environment specification—a configuration of the tyres such that the energy losses caused by them are minimized. Numerical tests on a set of optimization problems are presented to compare the performance of the algorithm developed with that of other existing algorithms.
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