Chapter 1 Introduction

Chapter Intro duction There is scarcely a mo dern journal whether of engineering economics management mathematics physics or the so cial sciences in which the concept optimization is missing from the sub ject index If one abstracts from all sp ecialist p oints of view the recurring problem is to select a b etter or b est Leibniz eventuallyheintro duced the term optimal alternativefromamonganumb er of p ossible states of aairs However if one were to followthehyp othesis of Leibniz as presented in his Theodicee that our world is the b est of all p ossible worlds one could justiably sink into passive fatalism There would b e nothing to improve or to optimize Biology esp ecially since Darwin has replaced the static world picture of Leibniz time by a dynamic one that of the more or less gradual development of the sp ecies culminat ing in the app earance of man Paleontology is providing an increasingly complete picture of organic evolution Socalled missing links rep eatedly turn out to b e not missing but rather hitherto undiscovered stages of this pro cess Very much older than the recogni tion that man is the result or b etter intermediate state of a meliorization pro cess is the seldomquestioned assumption that he is a p erfect end pro duct the pinnacle of cre ation Furthermore long b efore man conceived of himself as an active participantin the development of things he had unconsciously inuenced this evolution There can b e no doubt that his ability and eorts to make the environment meet his needs raised him ab ove other forms of life and have enabled him despite physical inferioritytondto hold and to extend his place in the worldso far at least As long as mankind has existed on our planet spaceship earth we together with other sp ecies havemutually inuenced and changed our environment Has this always b een done in the sense of meliorization In the French philosopher Voltaire dissatised with the conditions of his age was already taking up arms against Leibniz philosophical optimism and calling for conscious eort to change the state of aairs In the same waytodaywhenwe optimize we nd that we are b oth the sub ject and ob ject of the history of development In the desire to improve an ob ject a pro cess or a system Wilde and Beightler see an expression of the human striving for perfection Whether sucha lofty goal can b e attained dep ends on many conditions It is not p ossible to optimize when there is only one way to carry out a taskthen one has no alternative If it is not even known whether the problem at hand is soluble the Intro duction situation calls for an invention or discovery and not at that stage for any optimization But wherever two or more solutions exist and one must decide up on one of them one should cho ose the b est that is to say optimize Those indep endent features that distin guish the results from one another are called indep endent variables or parameters of the ob ject or system under consideration they may b e represented as binaryinteger other wise discrete or real values A rational decision between the real or imagined variants presupp oses a value judgement which requires a scale of values a quantitative criterion of merit according to which one solution can b e classied as b etter another as worse This dep endentvariable is usually called an objective function b ecause it dep ends on the ob jective of the systemthe goal to b e attained with itand is functionally related to the parameters There mayeven exist several ob jectives at the same timethe normal case in living systems where the mix of ob jectives also changes over time and may in fact b e induced by the actual course of the evolutionary paths themselves Sometimes the hardest part of optimization is to dene clearly an ob jective function For instance if several subgoals are aimed at a relativeweightmust b e attached to eachof the individual criteria If these are contradictory one only can hop e to nd a compromise on a tradeo subset of nondominated solutionsVariability and distinct order of merit are the unavoidable conditions of any optimization One may sometimes also think one has found the right ob jective for a subsystem only to realize later that in doing so one has provoked unwanted side eects the ramications of whichhaveworsened the disregarded total ob jective function We are just now exp eriencing hownarrowminded scales of value can steer us into dangerous plights and how it is sometimes necessary to consider the whole Earth as a system even if this is where dierences of opinion ab out value criteria are the greatest The second diculty in optimization particularly of multiparameter ob jectives or pro cesses lies in the choice or design of a suitable strategy for pro ceeding Even when the ob jective has b een suciently clearly dened indeed even when the functional dep endence on the indep endentvariables has b een mathematically or computationally formulated it often remains dicult enough if not completely imp ossible to nd the optimum esp ecially in the time available The uninitiated often think that it must b e an easy matter to solve a problem expressed in the language of mathematics that most exact of all sciences Far from it The problem of how to solve problems is unsolvedand mathematicians havebeenworking on it for centuries For giving exact answers to questions of extremal values and corresp onding p ositions or conditions we are indebted for example to the dierential and variational calculusofwhich the development in the th century is asso ciated with such illustrious names as Newton Euler Lagrange and Bernoulli These constitute the foundations of the present metho ds referred to as classical and of the further developments in the theory of optimization Still there is often a long way from the theorywhich is concerned with establishing necessary and sucient conditions for the existence of minima and maxima to the practice the determination of these most desirable conditions Practically signicant solutions of optimization problems rst b ecame p ossible with the arrival of large and fast programmable computers in the midth century Since then the o o d of publications on the sub ject of optimization has b een steadily rising in volume it is a Intro duction simple matter to collect several thousand published articles ab out optimization metho ds Even an interested party nds it dicult to keep pace nowadays with the development that is going on It seems far from b eing over for there still exists no allembracing theory of optimization nor is there anyuniversal metho d of solution Thus it is appropriate in Chapter to give a general survey of optimization problems and metho ds The sp ecial role of direct static nondiscrete and nonsto chastic parameter optimization emerges here for many of these metho ds can b e transferred to other elds the converse is less often p ossible In Chapter some of these strategies are presented in more depth principally those that extract the information they require only from values of the ob jective function thatistosay without recourse to analytic partial derivatives derivativefree methods Metho ds of a probabilistic nature are omitted here Metho ds which use chance as an aid to decision making are treated separately in Chapter In numerical optimizationchance is simulated deterministically by means of a pseudorandom numb er generator able to pro duce some kind of deterministic chaos only One of the random strategies proves to b e extremely promising It imitates in a highly simplied manner the mutationselection game of nature This concept a twomemb ered evolution strategy is formulated in a manner suitable for numerical optimization in Chap ter Section Following the hyp othesis put forward byRechenb erg that biological evolution is or p ossesses an esp ecially advantageous optimization pro cess and is there fore worthy of imitation an extended multimemb ered scheme that imitates the population principle of evolution is intro duced in Chapter Section It p ermits a more natural as well as more eective sp ecication of the step lengths than the two memberedscheme and actually invites the addition of further evolutionary principlessuch as for example sexual propagation and recombination An approximate theory of the rate of convergence can also b e set up for the evolution strategyinwhich only the b est of descendants of a generation b ecome parents of the following one A short excursion new to this edition intro duces nearly concurrentdevelopments that the author was unaware of when compiling his dissertation in the early s ie genetic algorithms simulated annealingandtabu search Chapter then makes a comparison of the evolution strategies with the direct search metho ds of zero rst and second order whichwere treated in detail in Chapter Since the predictivepower of theoretical pro ofs of convergence and statements of rates of convergence is limited to simple problem structures the comparison includes mainly numerical tests employing various mo del ob jective functions The results are evaluated from twopoints of view Eciency or sp eed of approach to the ob jective Eectivity or reliability under varying conditions The evolution strategies are highly successful in the test of eectivityor robustness Contrary to the widely held opinion that biological evolution is a very wasteful metho d of optimization the convergencerate test shows that in this resp ect to o the evolution metho ds can hold their own and are sometimes even more ecient than many purely deterministic metho ds The circle is closed

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