Subdivision Direction Selection in Interval Methods for Global Optimization

Csendes, Tibor; Ratz, Dietmar

doi:10.1137/s0036142995281528

Cited by 130 publications

(91 citation statements)

References 11 publications

(1 reference statement)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The choice of branching strategy (in our case which entry we choose to branch on in step 10 of Algorithm 1) can have a strong influence on the performance of branch-and-bound algorithms (see, for example, [2]). As will be evident from Proposition 4.3 in Section 4, in order to achieve theoretical convergence for a given precision, , it is necessary to branch on all (off-diagonal) interval entries.…”

Section: The Interval-matrix Branch-and-bound Algorithmmentioning

confidence: 99%

“…(4) Initialize L = {{[M], l, u}}, iter = 0. (5) while iter ≤ maxiters do (6) Choose the first entry, L 1 , from list L. [2], and u = L 1 [3]. (8) Delete L 1 from L. (9) if l < BUB then (10) Choose branching entry [m ij ], i = j.…”

Section: The Interval-matrix Branch-and-bound Algorithmmentioning

confidence: 99%

“…(24) end if (25) [2] being the second entry of the first sublist in L. (26) iter++. (27) if L is empty or BUB − BLB < then (28) Return BLB and BUB.…”

Section: The Interval-matrix Branch-and-bound Algorithmmentioning

confidence: 99%

“…Moreover, computing approximate solutions for the minimum and maximum eigenvalues of symmetric interval matrices can be NP-hard [8]. 2 …”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

An interval-matrix branch-and-bound algorithm for bounding eigenvalues

Nerantzis

Adjiman

2016

Optimization Methods and Software

View full text Add to dashboard Cite

We present and explore the behaviour of a branch-and-bound algorithm for calculating valid bounds on the kth largest eigenvalue of a symmetric interval matrix. Branching on the interval elements of the matrix takes place in conjunction with the application of Rohn's method (an interval extension of Weyl's theorem) in order to obtain valid outer bounds on the eigenvalues. Inner bounds are obtained with the use of two local search methods. The algorithm has the theoretical property that it provides bounds to any arbitrary precision > 0 (assuming infinite precision arithmetic) within finite time. In contrast with existing methods, bounds for each individual eigenvalue can be obtained even if its range overlaps with the ranges of other eigenvalues. Performance analysis is carried out through nine examples. In the first example, a comparison of the efficiency of the two local search methods is reported using 4000 randomly generated matrices. The eigenvalue bounding algorithm is then applied to five randomly generated matrices with overlapping eigenvalue ranges. Valid and sharp bounds are indeed identified given a sufficient number of iterations. Furthermore, most of the range reduction takes place in the first few steps of the algorithm so that significant benefits can be derived without full convergence. Finally, in the last three examples, the potential of the algorithm for use in algorithms to identify index-1 saddle points of nonlinear functions is demonstrated.

show abstract

Section: The Interval-matrix Branch-and-bound Algorithmmentioning

confidence: 99%

Section: The Interval-matrix Branch-and-bound Algorithmmentioning

confidence: 99%

“…(24) end if (25) [2] being the second entry of the first sublist in L. (26) iter++. (27) if L is empty or BUB − BLB < then (28) Return BLB and BUB.…”

Section: The Interval-matrix Branch-and-bound Algorithmmentioning

confidence: 99%

“…Moreover, computing approximate solutions for the minimum and maximum eigenvalues of symmetric interval matrices can be NP-hard [8]. 2 …”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An interval-matrix branch-and-bound algorithm for bounding eigenvalues

Nerantzis

Adjiman

2016

Optimization Methods and Software

View full text Add to dashboard Cite

show abstract

“…The investigation of some more sophisticated rules (see e.g. [3,15]) for circle packing problems can be a subject of a further study.…”

mentioning

confidence: 99%

A New Verified Optimization Technique for the "Packing Circles in a Unit Square" Problems

Markót¹,

Csendes²

2005

SIAM J. Optim.

View full text Add to dashboard Cite

Abstract. The paper presents a new verified optimization method for the problem of finding the densest packings of non-overlapping equal circles in a square. In order to provide reliable numerical results, the developed algorithm is based on interval analysis. As one of the most efficient parts of the algorithm, an interval-based version of a previous elimination procedure is introduced. This method represents the remaining areas still of interest as polygons fully calculated in a reliable way. Currently the most promising strategy of finding optimal circle packing configurations is to partition the original problem into subproblems. Still as a result of the highly increasing number of subproblems, earlier computer-aided methods were not able to solve problem instances where the number of circles was greater than 27. The present paper provides a carefully developed technique resolving this difficulty by eliminating large groups of subproblems together. As a demonstration of the capabilities of the new algorithm the problems of packing 28, 29, and 30 circles were solved within very tight tolerance values. Our verified procedure decreased the uncertainty in the location of the optimal packings by more than 700 orders of magnitude in all cases.Key words. interval arithmetic, branch-and-bound method, circle packing, optimality proof AMS subject classifications. 52C15, 52C26, 65G30, 65K05, 90C301. Introduction. The so-called 'optimal packing of equal circles into a square' problem class is an interesting part of geometrical optimization. Circle packing has several real-life applications, e.g. cutting out a given number of identical circle shaped objects from some kind of material with a minimal amount of waste. In addition, proving the optimality of a packing configuration is a serious theoretical challenge from both the mathematical and the computational points of view.The paper is organized as follows: in §2 we discuss the general definition of the problem and give a brief historical overview focusing mostly on those of the computeraided methods which form the basis of the present algorithm. Section 3 contains the basic definitions and properties of interval analysis. In §4 a general interval branchand-bound frame algorithm is introduced. After that, crucial parts of the B&B algorithm are discussed in detail, as those designed specifically for circle packing problems. As a specification step, §5 introduces a new elimination procedure based on a prior non-interval one known from the literature. In §6 the questions of finding global solutions are discussed and new methods for eliminating tile combinations are investigated. In §7 we propose an efficient way of handling occurrences of free (not fixed) circles in optimal packings. Finally, in §8 the results of the previous sections are applied by solving the packing problems of 28, 29, and 30 circles. The presented numerical results demonstrate how the algorithm works in the successive elimination steps.

show abstract

Lipschitz Global Optimization

Sergeyev

Kvasov

2011

Wiley Encyclopedia of Operations Research and Management Science

View full text Add to dashboard Cite

In this paper, a global optimization problem is considered where the objective function and constraints are multiextremal black‐box functions satisfying the Lipschitz condition over a hyperinterval and hard to evaluate. Such functions are frequently encountered in practice and that explains the great interest of researchers in the stated problem. Some known approaches to solving this problem are briefly described.

show abstract

Subdivision Direction Selection in Interval Methods for Global Optimization

Cited by 130 publications

References 11 publications

An interval-matrix branch-and-bound algorithm for bounding eigenvalues

An interval-matrix branch-and-bound algorithm for bounding eigenvalues

A New Verified Optimization Technique for the "Packing Circles in a Unit Square" Problems

Lipschitz Global Optimization

Contact Info

Product

Resources

About