Generalized McCormick relaxations

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

doi:10.1007/s10898-011-9664-7

Cited by 82 publications

(108 citation statements)

References 26 publications

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“…For linear dynamic systems, a method was developed in [61] which provides affine bounds in the parameter space via the solution of an auxiliary set of ODEs. This method was later extended to propagate affine bounds [62] as well as a pair of convex and concave bounds [58,60] for the solutions of parametric nonlinear ODEs based on differential inequalities and (generalized) McCormick relaxations [41,59]. The refinement of reachable set enclosures by accounting for a priori or physical information about a given system was also investigated in [57,61].…”

Section: Introductionmentioning

confidence: 99%

Unified framework for the propagation of continuous-time enclosures for parametric nonlinear ODEs

2014

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This paper presents a framework for constructing and analyzing enclosures of the reachable set of nonlinear ordinary differential equations (ODEs) using continuous-time setpropagation methods. The focus is on convex enclosures that can be characterized in terms of their support functions. A generalized differential inequality is introduced, whose solutions describe such support functions for a convex enclosure of the reachable set under mild conditions. It is shown that existing continuous-time bounding methods that are based on standard differential inequalities or ellipsoidal set propagation techniques can be recovered as special cases of this generalized differential inequality. A way of extending this approach for the construction of nonconvex enclosures is also described, which relies on Taylor models with convex remainder bounds. This unifying framework provides a means for analyzing the convergence properties of continuous-time enclosure methods. The enclosure techniques and convergence results are illustrated with numerical case studies throughout the paper, including a six-state dynamic model of anaerobic digestion.

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Section: Introductionmentioning

confidence: 99%

Unified framework for the propagation of continuous-time enclosures for parametric nonlinear ODEs

2014

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“…Similarly by feasibility and (x 1 , x 2 ) is also feasible in (38). It remains to show that the objective function of (38) is an underestimate of (37).…”

Section: Proof Of Propositionmentioning

confidence: 88%

“…Next we show that the optimization problem at the right hand of (38) is a relaxation of the optimization problem at the right hand of (37) and thus has a smaller optimal value. First we will show that any feasible point of (37) is also feasible in (38). Indeed, take any feasible ( x 1 , x 2 ).…”

Section: Proof Of Propositionmentioning

confidence: 98%

“…Scott and Barton [38] have observed thatḡ cv can be tightened by intersecting with the interval bounds. However, from the definition of g cv we have that g cv is at least as tight and in some cases tighter than the result by Scott and Barton.…”

Section: Corollary 5 Let G(z)mentioning

confidence: 99%

“…It remains to show that the objective function of (38) is an underestimate of (37). Take any (x 1 , x 2 ) which is feasible in (37).…”

Section: Proof Of Propositionmentioning

confidence: 99%

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Multivariate McCormick relaxations

Tsoukalas

Mitsos

2014

J Glob Optim

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McCormick (Math Prog 10(1): 1976) provides the framework for convex/concave relaxations of factorable functions, via rules for the product of functions and compositions of the form F • f , where F is a univariate function. Herein, the composition theorem is generalized to allow multivariate outer functions F, and theory for the propagation of subgradients is presented. The generalization interprets the McCormick relaxation approach as a decomposition method for the auxiliary variable method. In addition to extending the framework, the new result provides a tool for the proof of relaxations of specific functions. Moreover, a direct consequence is an improved relaxation for the product of two functions, at least as tight as McCormick's result, and often tighter. The result also allows the direct relaxation of multilinear products of functions. Furthermore, the composition result is applied to obtain improved convex underestimators for the minimum/maximum and the division of two functions for which current relaxations are often weak. These cases can be extended to allow composition of a variety of functions for which relaxations have been proposed.

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Worst‐case design of subsea production facilities using semi‐infinite programming

et al. 2014

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The problem of designing novel process systems for deployment in extreme and hostile environments is addressed. Specifically, the process system of interest is a subsea production facility for ultra deepwater oil and gas production. The costs associated with operational failures in deepwater environments are prohibitively high and, therefore, warrant the application of worst-case design strategies. That is, prior to the construction and deployment of a process, a certificate of robust feasibility is obtained for the proposed design. The concept of worst-case design is addressed by formulating the design feasibility problem as a semi-infinite optimization problem with implicit functions embedded. A basic model of a subsea production facility is presented for a case study of rigorous performance and safety verification. Relying on recent advances in global optimization of implicit functions and semi-infinite programming, the design feasibility problem is solved, demonstrating that this approach is effective in addressing the problem of worst-case design of novel process systems. Given a process model, and taking into account the uncertainty in the model and disturbances to the inputs of the system, do there exist control settings such that, at steady state, the physical system will always meet performance/safety specifications? This question will be formulated mathematically later and its application to subsea production facilities will be the primary focus of this article. In the following section, the subsea process system model will be presented and the case study will be set up. Model and Case StudyThe subsea separator is considered to be at the heart of subsea production facilities because it is the key process system for performing upstream phase separation as material is being produced from the wellhead. In the steadystate model presented here, it is considered that a threephase mixture of oil/water/gas is being sufficiently separated to allow for reinjection of the water back into the environment and the production of separate oil and gas Pointwise numerical simulation was performed using a Windows 7 machine with Intel Core2 Quad Q9450 CPU to study the behavior of the model over a range of uncertainty

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Generalized McCormick relaxations

Abstract: Convex relaxations, Global optimization, Optimal control, 49M20, 90C26,

Cited by 82 publications

References 26 publications

Unified framework for the propagation of continuous-time enclosures for parametric nonlinear ODEs

Unified framework for the propagation of continuous-time enclosures for parametric nonlinear ODEs

Multivariate McCormick relaxations

Worst‐case design of subsea production facilities using semi‐infinite programming

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