Convex and concave relaxations of implicit functions

Stuber, Matthew D.; Scott, Joseph K.; Barton, Paul I.

doi:10.1080/10556788.2014.924514

Cited by 39 publications

(96 citation statements)

References 25 publications

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“…The variables x (it) cannot be directly eliminated but rather the equations have to be given to the optimizer or addressed as implicit functions, see [30,38].…”

Section: Implicit Methodsmentioning

confidence: 99%

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Optimal deterministic algorithm generation

2018

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A formulation for the automated generation of algorithms via mathematical programming (optimization) is proposed. The formulation is based on the concept of optimizing within a parameterized family of algorithms, or equivalently a family of functions describing the algorithmic steps. The optimization variables are the parameters-within this family of algorithms-that encode algorithm design: the computational steps of which the selected algorithms consist. The objective function of the optimization problem encodes the merit function of the algorithm, e.g., the computational cost (possibly also including a cost component for memory requirements) of the algorithm execution. The constraints of the optimization problem ensure convergence of the algorithm, i.e., solution of the problem at hand. The formulation is described prototypically for algorithms used in solving nonlinear equations and in performing unconstrained optimization; the parametrized algorithm family considered is that of monomials in function and derivative evaluation (including negative powers). A prototype implementation in GAMS is provided along with illustrative results demonstrating cases for which well-known algorithms are shown to be optimal. The formulation is a 123J Glob Optim mixed-integer nonlinear program. To overcome the multimodality arising from nonconvexity in the optimization problem, a combination of brute force and general-purpose deterministic global algorithms is employed to guarantee the optimality of the algorithm devised. We then discuss several directions towards which this methodology can be extended, their scope and limitations.

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“…The variables x (it) cannot be directly eliminated but rather the equations have to be given to the optimizer or addressed as implicit functions, see [30,38].…”

Section: Implicit Methodsmentioning

confidence: 99%

“…Similarly for other values of ν 1 we have to calculate other steps via explicit matrix multiplication or via similar systems of equations. As aforementioned, the equations can be explicitly or implicitly written, see [30] and its extensions [38].…”

Section: Appendixmentioning

confidence: 99%

Optimal deterministic algorithm generation

2018

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“…15 This reduction of problem size can be taken to the extreme by eliminating all equality constraints along with suitable variables when using methods for relaxing implicit functions. 25,26 However, this is beyond the scope of this study. Formulation (RS) also reduces the number of variables for which bounds need to be provided (namely, the degrees of freedom d and the model variables x t and x m that are left to the optimizer).…”

Section: Optimization Formulationmentioning

confidence: 97%

“…These functions may, however, not contain general conditional statements or iterative procedures with an unknown number of iterations (such as Newton's method for solving nonlinear equation systems). While methods are recently being developed that can circumvent these limitations, [24][25][26] these are beyond the scope of this study. Instead, we seek to handle such instances through a suitable formulation of the optimization problem.…”

Section: Global Optimization Via Branch-and-boundmentioning

confidence: 99%

Deterministic global flowsheet optimization: Between equation‐oriented and sequential‐modular methods

Bongartz

Mitsos

2019

AIChE Journal

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Flowsheet optimization problems often exhibit multiple (suboptimal) local solutions. However, deterministic global optimization can get prohibitively expensive with increasing flowsheet size and has been mostly restricted to equation-oriented flowsheet models. In this work, we demonstrate how a reduced-space formulation for global optimization (a hybrid between equation-oriented and sequential-modular methods) can be applied to flash calculations as well as a process flowsheet combining flash units, reaction, and recycling, and investigate the influence of model formulation on solution time. Both for flash units using ideal or nonideal thermodynamics as well as for the flowsheet, the reduced-space formulation can be solved over an order of magnitude faster than a fully equation-oriented formulation. This demonstrates the potential of the approach even for flowsheets with implicit unit operations. However, the results also highlight the importance of identifying suitable model formulations at the unit operation level as well as the flowsheet level.

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“…3. The idea of reducing the search space, guided by additional information about the decision variables, is utilized also within constraint programming (Bessiere 2006;Stuber et al 2015).…”

Section: Previous Workmentioning

confidence: 99%

Efficient solution of many instances of a simulation-based optimization problem utilizing a partition of the decision space

et al. 2017

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This paper concerns the solution of a class of mathematical optimization problems with simulation-based objective functions. The decision variables are partitioned into two groups, referred to as variables and parameters, respectively, such that the objective function value is influenced more by the variables than by the parameters. We aim to solve this optimization problem for a large number of parameter settings in a computationally efficient way. The algorithm developed uses surrogate models of the objective function for a selection of parameter settings, for each of which it computes an approximately optimal solution over the domain of the variables. Then, approximate optimal solutions for other parameter settings are computed through a weighting of the surrogate models without requiring additional expensive function evaluations. We have tested the algorithm's performance on a set of global optimization problems differing with respect to both mathematical properties and numbers of variables and parameters. Our results show that it outperforms a standard and often applied approach based on a surrogate model of the objective function over the complete space of variables and parameters.

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Convex and concave relaxations of implicit functions

Cited by 39 publications

References 25 publications

Optimal deterministic algorithm generation

Optimal deterministic algorithm generation

Deterministic global flowsheet optimization: Between equation‐oriented and sequential‐modular methods

Efficient solution of many instances of a simulation-based optimization problem utilizing a partition of the decision space

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