## Simulated Annealing Algorithms For Continuous Global Optimization (2000)

Citations: | 32 - 1 self |

### BibTeX

@MISC{Locatelli00simulatedannealing,

author = {M. Locatelli},

title = {Simulated Annealing Algorithms For Continuous Global Optimization},

year = {2000}

}

### Years of Citing Articles

### OpenURL

### Abstract

INTRODUCTION In this paper we consider Simulated Annealing algorithms (SA in what follows) applied to continuous global optimization problems, i.e. problems with the following form f = min x2X f(x); (1.1) where X ` ! n is a continuous domain, often assumed to be compact, which, combined with the continuity or lower semicontinuity of f , guarantees the existence of the minimum value f . SA algorithms are based on an analogy with a physical phenomenon: while at high temperatures the molecules in a liquid move freely, if the temperature is slowly decreased the thermal mobility of the molecules is lost and they form a pure crystal which also corresponds to a state of minimum energy. If the temperature is decreased too quickly (the so called quenching) a liquid metal rather ends up in a polycrystalline or amorphous state with

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Citation Context ...ollowing. Step 1. Let Y0∈X be a given starting point, let z0G{(Y0, f(Y0))}, f *0 Gf(Y0), and kG0. Step 2. Sample a point XkC1 from a distribution D(·; zk). Step 3. Sample a uniform random number p in =-=[0, 1]-=- and set YkC1G� XkC1, if pYA(Yk,XkC1,tk), Yk, otherwise, where A is a function with values in [0, 1] and tk is a parameter called the temperature at iteration k. Step 4. Set zkC1Gzk∪{(XkC1, f(XkC1))};... |

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Citation Context ...here D(·, ·) is a Markov kernel, absolutely continuous with respect to the Lebesgue measure m and with density β bounded away from zero, i.e., D(x, B)G� β(x, y) dy and inf β(x, y)GρH0, B x,y∈X ∀B∈B , =-=(2)-=- where B is a σ-algebra over X. (A2) For any G⊆X open, D(x, G) is continuous in x. (A3) For any initial state Y0 and any initial temperature t0, (A4) tkGU(zk)→0, (3) with probability 1. X is a compact... |

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Citation Context ...ys implied by Assumption 3.8. Indeed, for any x∈B22¯ \B2/2, the probability in (12) can be split in the following sum P[ f(YkC1)Xf(Yk)Cuk�YkGx, RkGrk, zk] CP[ f(Yk)Ff(YkC1)Ff(Yk)Cuk�YkGx, RkGrk, zk]. =-=(13)-=-130 JOTA: VOL. 104, NO. 1, JANUARY 2000 By observing the Metropolis acceptance function (1), it follows that an upper bound for the first probability in (13) is the value exp{−(uk�ck)}, while from As... |

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Citation Context ...point XkC1 is equivalent to the uniform distribution over S(Yk, Rk)∩X; i.e., there exist constants g, GH0, gYG such that g[m(B)�m(S(Yk, Rk)∩X)]YD(B; zk)YG[m(B)�m(S(Yk, Rk)∩X)], ∀B⊆S(Yk, Rk)∩X, ∀Yk∈X. =-=(7)-=- The second assumption introduces restrictions on the objective function f and the feasible set X. Assumption 3.2. The feasible set X is compact, convex, and fulldimensional, and the objective functio... |

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Citation Context ...nd inf β(x, y)GρH0, B x,y∈X ∀B∈B , (2) where B is a σ-algebra over X. (A2) For any G⊆X open, D(x, G) is continuous in x. (A3) For any initial state Y0 and any initial temperature t0, (A4) tkGU(zk)→0, =-=(3)-=- with probability 1. X is a compact set, f is a continuous function, and m(B2)H0, ∀2H0. Under the above assumptions, it has been proved that lim P[Yk∈B2]G1, k→S ∀2H0. Note that (3) is much less restri... |

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Citation Context ...following result can be proved.128 JOTA: VOL. 104, NO. 1, JANUARY 2000 Theorem 3.1. let Assumptions 3.1–3.3 hold. Then, Let tk and Rk be given by (4) and (6), respectively, and P[∃k: Yk∈B2]G1, ∀2H0. =-=(10)-=- Proof. See Theorem 2 in Ref. 11. � Basically, Theorem 3.1 states that, under the given assumptions, the record value f *k converges to the global optimum value f * with probability 1. However, Assump... |

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Citation Context ...of accepting an ascent step decreases to 0 as the iteration counter k increases. Assumption 3.7. There exists a sequence bkGbk(2)→0 such that P[ f(YkC1)Hf(Yk)�YkGx, RkGrk, zk]Ybk, ∀x∈B22¯ \B2/2, ∀zk. =-=(12)-=- While Assumption 3.7 is what we need for the proof of (11), we would like to introduce an alternative assumption, which is less general than Assumption 3.7, but gives a better indication about how {c... |

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Citation Context ...the assumption is the following. Assumption 3.3. The sequences {ck} and {rk} decrease to 0 slowly enough; specifically, the following condition must be satisfied: ∑ S ψ sq j exp{A[sqj ∆Fqj�cqjC1]}GS, =-=(9)-=- jG1 where ψ∈(0, 1) is a given constant. While condition (9) is quite general, it is quite difficult to appreciate its dependency on the two sequences {ck} and {rk}. In particular, while the sequence ... |

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Citation Context ...rges with probability 1 to the global optimum. Theorem 4.2. Let 2˜ be the same as in Theorem 4.1. If Assumptions 3.1–3.7 hold, and if in (4) and (6) 2¯ ∈(0, 2˜ ] is chosen, then lim P[Xk∈B2]G1, ∀2H0. =-=(14)-=- k→S Proof. See Theorem 4 in Ref. 11. � As already remarked in the previous section, a drawback of the above result is that generally it is not possible to know in advance a suitable value for 2¯. How... |

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Citation Context ...phere for the distribution D(·; zk); i.e., we should have RkXδ 2H0, if f(Yk)Af *k H2¯, (6a) RkGrk, otherwise, (6b) where {rk} is a deterministic nonincreasing sequence converging to 0. The meaning of =-=(6)-=- is that, if we are in a point whose function value is poor with respect to the current record, then we guarantee the possibility of performing steps XkC1AYk whose length is bounded away from 0, while... |

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Citation Context ...the case of finitely many global optima is trivial). Note that the continuity of f and the compactness of X imply that f is uniformly continuous over X; i.e., ∀δH0, ∃γGγ (δ)H0:d(x,y)Yγ⇒�f(x)Af(x)�Yδ. =-=(8)-=-JOTA: VOL. 104, NO. 1, JANUARY 2000 127 Next, we introduce an assumption about the speed of convergence to 0 for the sequences {ck} and {rk}. First, we need to introduce some notation. Let skGmin� s:... |

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