The B<scp>oxstep</scp> Method for Large-Scale Optimization

Marsten, Roy E.; Hogan, William W.; Blankenship, J. W.

doi:10.1287/opre.23.3.389

Cited by 139 publications

(11 citation statements)

References 9 publications

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“…Therefore, to gain suitable Lagrangean multipliers is a lot time consuming. For removing this problem the trust region method which is proposed by Marsten [13] is used. In this method, better updating of Lagrangean multipliers can be expected while the convergence properties are established.…”

Section: Lagrangean Relaxation Methodsmentioning

confidence: 99%

A New Combination of Lagrangean Relaxation, Dantzig-Wolfe Decomposition and Benders Decomposition Methods for Exact Solution of the Mixed Integer Programming Problems

Mohammadi

Khorram

2019

Comm. Math. & Appl.

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The combinational technique of cross decomposition is a suitable one for exact solution of the mixed integer programming problems which uses simultaneously the advantages of Lagrangean relaxation, Dantzig-Wolfe decomposition and Benders decomposition methods for Minimization problem that each reinforces one another. The basic idea for this technique is the generation of suitable upper and lower bounds for the optimal value of the original problem at each iteration. In this paper, new cross decomposition algorithm, with the combination of Lagrangean relaxation method (the combination of three concepts of cutting-plane, sub-gradient and trust region), Dantzig-Wolfe decomposition and Benders decomposition methods are used in order to reinforce bounds and to speed up convergence. By increasing the problem scale and regarding the use of the Lagrangean relaxation method in this technique, the lower bound with more strength and efficacy, and by the aid of Dantzig-Wolfe decomposition method more suitable upper bound (if exists) and furthermore less number of iterations for achieving optimal solution is obtained. The convergence of this technique regarding the convergence of Benders decomposition method in finite iteration numbers is guaranteed.

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Section: Lagrangean Relaxation Methodsmentioning

confidence: 99%

A New Combination of Lagrangean Relaxation, Dantzig-Wolfe Decomposition and Benders Decomposition Methods for Exact Solution of the Mixed Integer Programming Problems

Mohammadi

Khorram

2019

Comm. Math. & Appl.

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“…In practice, the cutting plane method sometimes cannot ensure the convergence speed and many iterations are required. Some refinement methods such as the boxstep method and the volume algorithm are proposed. Mouret et al also proposed a modified method to update the Lagrange multipliers .…”

Section: Solution Frameworkmentioning

confidence: 99%

“…In the continuous time formation of the cracking furnace system, the sequential constraints for the starting and ending times of global continuous time slot t are presented in Equations (35)(36)(37)(38). Equations (35)(36)(37) model the time sequence between starting time Ts j,t and ending time Tf j,t . Equation (38) gives the total planning horizon for the global continuous time slots.…”

Section: Furnace Time Sequence Constraintsmentioning

confidence: 99%

Integrated short‐term scheduling and production planning in an ethylene plant based on Lagrangian decomposition

Wang

Feng

et al. 2016

Can J Chem Eng

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A cracking furnace, which is the major unit involved in ethylene production, produces various petrochemical products ranging from ethylene to pitch and determines the production yield of an ethylene plant. It is essential to consider the operational performances of the cracking furnaces and to incorporate them into the whole plant's planning problem. Facing this challenge, this paper developed an optimization model, which integrates the scheduling problem of upstream cracking furnaces and the operational planning of the downstream units under a synchronized global time scale. Moreover, a modified Lagrangian decomposition algorithm is proposed for solving the large‐scale mixed integer nonlinear optimization problem. An industrial case study demonstrates the feasibility of the integrated model and the effectiveness of the solution algorithm.

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“…M in (1)), and that D τ should be smooth at 0 (that is, D * τ should be strictly convex at zero) so that convergence can be attained even for fixed or vanishing τ [12], and a primal optimal solution can be efficiently recovered [5]. A first attempt in this direction has been made in [10], where a 3-piecewise function is proposed that somewhat merges [26] with [20]: a linear stabilization is used, but only outside of a small region where violation of the constraints is not penalized. However, this may suffer from the same shortcomings of the Boxstep method, in that the penalties must be high to ensure boundedness (and, more in general, to avoid the same unstable behavior as CG), so only small moves in the non-penalized region may ultimately be performed, slowing down convergence.…”

Section: Stabilizing Functionsmentioning

confidence: 99%

“…The stability center is changed if a "sufficiently better" dual point is found. A variety of stabilized algorithms of this kind has been proposed [19,20,23,26,32], and a deeper theoretical understanding of the underlying principles [12,18,33] has been achieved over time; we especially refer the interested reader to [24].…”

Section: Introductionmentioning

confidence: 99%

On the choice of explicit stabilizing terms in column generation

Amor

Desrosiers

Frangioni

2009

Discrete Applied Mathematics

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Column generation algorithms are instrumental in many areas of applied optimization, where linear programs with an enormous number of columns need to be solved. Although successfully employed in many applications, these approaches suffer from well-known instability issues that somewhat limit their efficiency. Building on the theory developed for nondifferentiable optimization algorithms, a large class of stabilized column generation algorithms can be defined which avoid the instability issues by using an explicit stabilizing term in the dual; this amounts at considering a (generalized) augmented Lagrangian of the primal master problem. Since the theory allows for a great degree of flexibility in the choice and in the management of the stabilizing term, one can use piecewise-linear or quadratic functions that can be efficiently dealt with off-the-shelf solvers. The effectiveness in practice of this approach is demonstrated by extensive computational experiments on large-scale Vehicle and Crew Scheduling problems. Also, the results of a detailed computational study on the impact of the different choices in the stabilization term (shape of the function, parameters), and their relationships with the quality of the initial dual estimates, on the overall effectiveness of the approach are reported, providing practical guidelines for selecting the most appropriate variant in different situations.

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The Boxstep Method for Large-Scale Optimization

Abstract: A new strategy is presented for large-scale optimization. The Boxstep method creates an algorithmic continuum between feasible-directions methods and cutting-plane methods. Several specific applications are described and computational results are reported.

Cited by 139 publications

References 9 publications

A New Combination of Lagrangean Relaxation, Dantzig-Wolfe Decomposition and Benders Decomposition Methods for Exact Solution of the Mixed Integer Programming Problems

A New Combination of Lagrangean Relaxation, Dantzig-Wolfe Decomposition and Benders Decomposition Methods for Exact Solution of the Mixed Integer Programming Problems

Integrated short‐term scheduling and production planning in an ethylene plant based on Lagrangian decomposition

On the choice of explicit stabilizing terms in column generation

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