The cluster problem in constrained global optimization

Kannan, Rohit; Barton, Paul I.

doi:10.1007/s10898-017-0531-z

Cited by 11 publications

(23 citation statements)

References 25 publications

(66 reference statements)

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“…The following notion of convergence for schemes of relaxations originates in Bompadre et al 27 Note that the constants c and s in Definition 4 may depend on � S, but not on S. First-order convergence is necessary for finite termination of spatial branch-and-bound algorithms, while second-order convergence is known to be critical for efficient branch-and-bound because it can eliminate the cluster effect, which refers to the accumulation of a large number of branch-and-bound nodes near a global solution. 1,27,28 Moreover, note that pointwise convergence of order c is stronger than Hausdorff convergence of order c (see Theorem 1 in Bompadre et al 27 ).…”

Section: Preliminariesmentioning

confidence: 98%

“…In fact, the second-order convergence rate established here is known to be critical for avoiding the so-called cluster effect in branch-and-bound, and hence avoiding exponential run-time in practice. 28 Third, although refining the partition of � X is required for convergence, valid relaxations on X can be obtained using any partition of � X, no matter how coarse. Thus, the partition of � X can be adaptively refined as the branch-andbound search proceeds.…”

Section: Introductionmentioning

confidence: 99%

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Convex Relaxations for Global Optimization Under Uncertainty Described by Continuous Random Variables

Shao¹,

Scott²

2017

Preprint

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This article considers nonconvex global optimization problems subject to uncertainties described by continuous random variables. Such problems arise in chemical process design, renewable energy systems, stochastic model predictive control, and many other applications. Here, we restrict our attention to problems with expected-value objectives and no recourse decisions. In principle, such problems can be solved globally using spatial branch-and-bound. However, branch-and-bound requires the ability to bound the optimal objective value on subintervals of the search space, and existing techniques are not generally applicable because expected-value objectives often cannot be written in closedform. To address this, this article presents a new method for computing convex and concave relaxations of nonconvex expected-value functions, which can be used to obtain rigorous bounds for use in branch-and-bound. Furthermore, these relaxations obey a second-order pointwise convergence property, which is sufficient for finite termination of branchand-bound under standard assumptions. Empirical results are shown for three simple examples. V C 2017 American Institute of Chemical Engineers AIChE J, 00: 000-000, 2018

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Section: Preliminariesmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Convex Relaxations for Global Optimization Under Uncertainty Described by Continuous Random Variables

Shao¹,

Scott²

2017

Preprint

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show abstract

“…Note that the constants γ and τ in Definition 4 may depend on

\overset{S}{true}

, but not on S . First‐order convergence is necessary for finite termination of spatial branch‐and‐bound algorithms, while second‐order convergence is known to be critical for efficient branch‐and‐bound because it can eliminate the cluster effect , which refers to the accumulation of a large number of branch‐and‐bound nodes near a global solution . Moreover, note that pointwise convergence of order γ is stronger than Hausdorff convergence of order γ (see Theorem 1 in Bompadre et al).…”

Section: Preliminariesmentioning

confidence: 99%

“…This is in contrast to both stochastic branch‐and‐bound and sample‐average approximation, which only converge in the limit of infinite sampling. In fact, the second‐order convergence rate established here is known to be critical for avoiding the so‐called cluster effect in branch‐and‐bound, and hence avoiding exponential run‐time in practice . Third, although refining the partition of

\overset{Ω}{true}

is required for convergence, valid relaxations on X can be obtained using any partition of

\overset{Ω}{true}

, no matter how coarse.…”

Section: Introductionmentioning

confidence: 96%

Convex relaxations for global optimization under uncertainty described by continuous random variables

Shao

Scott

2018

AIChE Journal

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This article considers nonconvex global optimization problems subject to uncertainties described by continuous random variables. Such problems arise in chemical process design, renewable energy systems, stochastic model predictive control, etc. Here, we restrict our attention to problems with expected-value objectives and no recourse decisions. In principle, such problems can be solved globally using spatial branch-and-bound (B&B). However, B&B requires the ability to bound the optimal objective value on subintervals of the search space, and existing techniques are not generally applicable because expected-value objectives often cannot be written in closed-form. To address this, this article presents a new method for computing convex and concave relaxations of nonconvex expected-value functions, which can be used to obtain rigorous bounds for use in B&B. Furthermore, these relaxations obey a secondorder pointwise convergence property, which is sufficient for finite termination of B&B under standard assumptions. Empirical results are shown for three simple examples.

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“…Establishing that a scheme of relaxations is at least second-order Hausdorff convergent is important from many viewpoints, notably in mitigating the so-called cluster effect in unconstrained global optimization [7,39]. Recently, the authors of this work have analyzed the cluster problem for constrained global optimization and determined that, under certain conditions, first-order convergence of the lower bounding scheme may be sufficient to avoid the cluster problem at constrained minima [15]. However, an analysis of convergence order for constrained problems is currently lacking.…”

Section: Introductionmentioning

confidence: 99%

Convergence-order analysis of branch-and-bound algorithms for constrained problems

Kannan

Barton

2017

J Glob Optim

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The performance of branch-and-bound algorithms for deterministic global optimization is strongly dependent on the ability to construct tight and rapidly convergent schemes of lower bounds. One metric of the efficiency of a branch-and-bound algorithm is the convergence order of its bounding scheme. This article develops a notion of convergence order for lower bounding schemes for constrained problems, and defines the convergence order of convex relaxation-based and Lagrangian dual-based lower bounding schemes. It is shown that full-space convex relaxation-based lower bounding schemes can achieve first-order convergence under mild assumptions. Furthermore, such schemes can achieve second-order convergence at KKT points, at Slater points, and at infeasible points when second-order pointwise convergent schemes of relaxations are used. Lagrangian dual-based full-space lower bounding schemes are shown to have at least as high a convergence order as convex relaxation-based full-space lower bounding schemes. Additionally, it is shown that Lagrangian dual-based full-space lower bounding schemes achieve first-order convergence even when the dual problem is not solved to optimality. The convergence order of some widely-applicable reduced-space lower bounding schemes is also analyzed, and it is shown that such schemes can achieve first-order convergence under suitable assumptions. Furthermore, such schemes can achieve second-order convergence at KKT points, at unconstrained points in the reduced-space, and at infeasible points under suitable assumptions when the problem exhibits a specific separable structure. The importance of constraint propagation techniques in boosting the convergence order of reduced-space lower bounding schemes (and helping mitigate clustering in the process) for problems which do not possess such a structure is demonstrated. Keywords Global optimization • Constrained optimization • Convergence order • Convex relaxation • Lagrangian dual • Branch-and-bound • Lower bounding scheme • Reduced-space Mathematics Subject Classification (2010) 49M20 • 49M29 • 49M37 • 49N15 • 65K05 • 68Q25 • 90C26 The authors gratefully acknowledge financial support from BP. This work was conducted as a part of the BP-MIT conversion research program.

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The cluster problem in constrained global optimization

Cited by 11 publications

References 25 publications

Convex Relaxations for Global Optimization Under Uncertainty Described by Continuous Random Variables

Convex Relaxations for Global Optimization Under Uncertainty Described by Continuous Random Variables

Convex relaxations for global optimization under uncertainty described by continuous random variables

Convergence-order analysis of branch-and-bound algorithms for constrained problems

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