Multivariate McCormick relaxations

Tsoukalas, Angelos; Mitsos, Alexander

doi:10.1007/s10898-014-0176-0

Cited by 70 publications

(98 citation statements)

References 49 publications

(55 reference statements)

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“…Instead, it might be possible to directly create multivariable Chebyshev approximations, for instance by following a similar approach as in chebfun2 [56]. This idea is also similar to the recent work on multivariate McCormick relaxations [59], where instead of decomposing the factorable function down to binary sums, products and univariate compositions, it would be possible to directly bound some multivariate terms. …”

Section: Discussionmentioning

confidence: 97%

Chebyshev model arithmetic for factorable functions

et al. 2016

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This article presents an arithmetic for the computation of Chebyshev models for factorable functions and an analysis of their convergence properties. Similar to Taylor models, Chebyshev models consist of a pair of a multivariate polynomial approximating the factorable function and an interval remainder term bounding the actual gap with this polynomial approximant. Propagation rules and local convergence bounds are established for the addition, multiplication and composition operations with Chebyshev models. The global convergence of this arithmetic as the polynomial expansion order increases is also discussed. A generic implementation of Chebyshev model arithmetic is available in the library MC++. It is shown through several numerical case studies that Chebyshev models provide tighter bounds than their Taylor model counterparts, but this comes at the price of extra computational burden.

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Section: Discussionmentioning

confidence: 97%

Chebyshev model arithmetic for factorable functions

et al. 2016

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“…We thus have to either solve the problems as a DNLP or reformulate it using auxiliary integer variables. In principle introducing auxiliary variables seems overcomplicated, but note that only recently [42] convex relaxations for the min operator were proposed.…”

Section: Methodsmentioning

confidence: 99%

Global optimization of generalized semi-infinite programs via restriction of the right hand side

Mitsos

Tsoukalas

2014

J Glob Optim

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The algorithm proposed in Mitsos (Optimization 60(10-11):1291(Optimization 60(10-11): -1308(Optimization 60(10-11): , 2011 for the global optimization of semi-infinite programs is extended to the global optimization of generalized semi-infinite programs. No convexity or concavity assumptions are made. The algorithm employs convergent lower and upper bounds which are based on regular (in general nonconvex) nonlinear programs (NLP) solved by a (black-box) deterministic global NLP solver. The lower bounding procedure is based on a discretization of the lower-level host set; the set is populated with Slater points of the lower-level program that result in constraint violations of prior upper-level points visited by the lower bounding procedure. The purpose of the lower bounding procedure is only to generate a certificate of optimality; in trivial cases it can also generate a global solution point. The upper bounding procedure generates candidate optimal points; it is based on an approximation of the feasible set using a discrete restriction of the lower-level feasible set and a restriction of the right-hand side constraints (both lower and upper level). Under relatively mild assumptions, the algorithm is shown to converge finitely to a truly feasible point which is approximately optimal as established from the lower bound. Test cases from the literature are solved and the algorithm is shown to be computationally efficient.

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“…Moreover, to estimate the cost of such algorithms the results of ongoing analysis in the spirit of automatic differentiation will be required [10,16] and the cost does not only depend on the values of the derivatives at a point. For instance in αBB [1,27] one needs to estimate the eigenvalues of the Hessian for the domain; in McCormick relaxations [28,42] the cost depends on how many times the composition theorem has to be applied. Recall also the discussion on cost for some local algorithms such as line-search methods.…”

Section: Global Optimizationmentioning

confidence: 99%

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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Multivariate McCormick relaxations

Cited by 70 publications

References 49 publications

Chebyshev model arithmetic for factorable functions

Chebyshev model arithmetic for factorable functions

Global optimization of generalized semi-infinite programs via restriction of the right hand side

Optimal deterministic algorithm generation

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