2 discretized variales approach those of the original continuous variales. Such an assumption is valid when continuous variales are represented as oat

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1 Chapter 1 CONSTRAINED GENETIC ALGORITHMS AND THEIR APPLICATIONS IN NONLINEAR CONSTRAINED OPTIMIZATION Benjamin W. Wah and Yi-Xin Chen Department of Electrical and Computer Engineering and the Coordinated Science Laoratory University of Illinois, Urana-Champaign Urana, IL 61801, USA Astract This chapter presents a framework that unies various search mechanisms for solving constrained nonlinear programming (NLP) prolems. These prolems are characterized y functions that are not necessarily dierentiale and continuous. Our proposed framework is ased on the rst-order necessary and sucient condition developed for constrained local minimization in discrete space that shows the equivalence etween discrete-neighorhood saddle points and constrained local minima. To look for discrete-neighorhood saddle points, we formulate a discrete constrained NLP in an augmented Lagrangian function and study various mechanisms for performing ascents of the augmented function in the original-variale suspace and descents in the Lagrange-multiplier suspace. Our results show that CSAGA, a comined constrained simulated annealing and genetic algorithm, performs well when using crossovers, mutations, and annealing to generate trial points. Finally, we apply iterative deepening to determine the optimal numer of generations in CSAGA and show that its performance is roust with respect to changes in population size. 1. Introduction Many engineering applications can e formulated as constrained nonlinear programming prolems (NLPs). Examples include production planning, computer integrated manufacturing, chemical control processing, and structure optimization. These applications can e solved y existing methods if they are specied in well-dened formulae that are dierentiale and continuous. However, only special cases can e solved when they do not satisfy the required assumptions. For instance, sequential quadratic programming cannot handle prolems whose ojective and constraint functions are not dierentiale or whose variales are discrete or mixed. Since many applications involving optimization may e formulated y non-dierentiale functions with discrete or mixed-integer variales, it is important to develop new methods for handling these optimization prolems. The study of algorithms for solving a disparity of constrained optimization prolems is dicult unless the prolems can e represented in a unied way. In this chapter we assume that continuous variales are rst discretized into discrete variales in such a way that the values of functions using Research supported y National Aeronautics and Space Administration Contract NAS

2 2 discretized variales approach those of the original continuous variales. Such an assumption is valid when continuous variales are represented as oating-point numers and when the range of variales is small (say etween 10 5 and 10 5 ). Intuitively, if discretization is ne enough, then solutions found in discretized space are fairly good approximations to the original solutions. The accuracy of solutions found in discretized prolems has een studied elsewhere [16]. Based on discretization, continuous and mixed-integer constrained NLPs can e represented as discrete constrained NLPs as follows: 1 minimize f(x) (1.1) suject to g(x) 0 x = [x 1 ; : : : ; x n ] T is a vector h(x) = 0 of ounded discrete variales. Here, f(x) is a lower-ounded ojective function, g(x) = [g 1 (x); ; g k (x)] T is a vector of k inequality constraints, h(x) = [h 1 (x); ; h m (x)] T is a vector of m equality constraints. Functions f(x), g(x), and h(x) are not necessarily dierentiale and can e either linear or nonlinear, continuous or discrete, and analytic or procedural. Without loss of generality, we consider only minimization prolems. Solutions to (1.1) cannot e characterized in ways similar to those of prolems with dierentiale functions and continuous variales. In the latter class of prolems, solutions are dened with respect to neighorhoods of open spheres with radius approaching zero asymptotically. Such a concept does not exist in prolems with discrete variales. Let X e the Cartesian product of the discrete domains of all variales in x. To characterize solutions sought in discrete space, we dene the following concepts on neighorhoods and constrained solutions in discrete space: Denition 1. N dn (x), the discrete neighorhood [1] of point x 2 X is a nite user-dened set of points fx 0 2 Xg such that x 0 2 N dn (x) () x 2 N dn (x 0 ), and that for any y 1 ; y k 2 X, it is possile to nd a nite sequence of points in X, y 1 ; ; y k, such that y i+1 2 N dn (y i ) for i = 1; k 1. Denition 2. Point x 2 X is called a constrained local minimum in discrete neighorhood (CLM dn ) if it satises two conditions: a) x is feasile, and ) f(x) f(x 0 ), for all feasile x 0 2 N dn (x). Denition 3. Point x 2 X is called a constrained gloal minimum in discrete neighorhood (CGM dn ) i a) x is feasile, and ) for every feasile point x 0 2 X, f(x 0 ) f(x). The set of all CGM dn is X opt. According to our denitions, a CGM dn must also e a CLM dn. In a similar way, there are denitions on continuous-neighorhood constrained local minima (CLM cn ) and constrained gloal minima (CGM cn ). We have shown earlier [14] that the necessary and sucient condition for a point to e a CLM dn is that it satises the discrete-neighorhood saddle-point condition (Section 2.1). We have also extended simulated annealing (SA) [13] and greedy search [14] to look for discrete-neighorhood saddle points SP dn (Section 2.2). At the same time, new prolem-dependent constraint-handling heuristics have een developed in the GA community to handle nonlinear constraints [11] (Section 2.3). Up to now, there is no clear understanding on how to unify these algorithms into one that can e applied to nd CGM dn for a wide range of prolems. 1 For two vectors v and w of the same numer of elements, v w means that each element of v is not less than the corresponding element of w. v w can e dened similarly. 0, when compared to a vector, stands for a null vector.

3 Constrained Genetic Algorithms and their Applications in Nonlinear Constrained Optimization 2 3 P R (N g P ) N g P N gp P R (N gp ) N g P a) P R(N gp ) approaches one asymptotically ) Existence of asolute minimum N optp in N gp P R (N gp ) Figure 1.1. An example showing the application of CSAGA with P = 3 to solve a discretized version of G1 [11] (N optp 2000). Based on our previous work, our goal in this chapter is to develop an eective framework that unies SA, GA, and greedy search for nding CGM dn. In particular, we propose constrained genetic algorithm (CGA) and comined constrained SA and GA (CSAGA) that look for SP dn. We also study algorithms with the optimal average completion time for nding a CGM dn. The algorithms studied in this chapter are all stochastic searches that proe a search space in a random order, where a proe is a neighoring point examined y an algorithm, independent of whether it is accepted or not. Assuming p j to e the proaility that an algorithm nds a CGM dn in its j th proe and a simplistic assumption that all proes are independent, the performance of one run of such an algorithm can e characterized y N, the numer of proes made (or CPU time taken), and P R (N), the reachaility proaility that a CGM dn is hit in any of the N proes: P R (N) = 1 NY j=1 (1 p j ); where N 0: (1.2) Reachaility can e maintained y reporting the est solution found y the algorithm when it stops. As an example, Figure 1.1a plots P R (N g P ) when CSAGA (see Section 3.2) was run under various numer of generations N g and xed population size P = 3 (where N = N g P ). The graph shows that P R (N g P ) approaches one asymptotically as N g P is increased. Although it is hard to estimate the value of P R (N) when a test prolem is solved y an algorithm, we can always improve the chance of nding a solution y running the same algorithm multiple times, each with N proes, from random starting points. Given P R (N) for one run of the algorithm and that all runs are independent, the expected total numer of proes to nd a CGM dn is: 1X j=1 P R (N)(1 P R (N)) j1 N j = Figure 1.1 plots (1.3) ased on P R (N g P ) in Figure 1.1a. minimizes (1.3) ecause P R (0) = 0, lim N!1 P R (N ) = 1, N P R (N) N P R (N) : (1.3) N P R (N)! 1 as N! 1. The curve in Figure 1.1 illustrates this ehavior. In general, there exists N opt that is ounded elow y zero, and

4 4 Based on the existence of N opt, we present in Section 3.3 search strategies in CGA and in CSAGA that minimize (1.3) in nding a CGM dn. Finally, Section 4 compares the performance of our algorithms. 2. Previous Work In this section, we rst summarize the theory of Lagrange multipliers applied to solve (1.1) and two algorithms developed ased on the theory. We then descrie existing work in GA for solving constrained NLPs Theory of Lagrange multipliers for solving discrete constrained NLPs Dene a discrete equality-constrained NLP as follows [14, 16]: min x f(x) x is a vector of ounded (1.4) suject to h(x) = 0 discrete variales, A generalized discrete augmented Lagrangian function of (1.4) is dened as follows [14]: L d (x; ) = f(x) + T H (h(x)) jjh(x)jj2 ; (1.5) where H is a non-negative continuous transformation function satisfying H(y) 0, H(y) = 0 i y = 0, and = [ 1 ; ; m ] T is a vector of Lagrange multipliers. Function H is easy to design; examples include H(h(x)) = [jh 1 (x)j; ; jh m (x)j] T and H(h(x)) = [max(h 1 (x); 0); ; max(h m (x); 0)] T. Note that these transformations are not used in Lagrangemultiplier methods in continuous space ecause they are not dierentiale at H(h(x)) = 0. However, they do not pose prolems here ecause we do not require their dierentiaility. Similar transformations can e used to transform inequality constraint g j (x) 0 into equivalent equality constraint max(g j (x); 0) = 0. Hence, we only consider prolems with equality constraints from here on. We dene a discrete-neighorhood saddle point SP dn (x ; ) with the following property: L d (x ; ) L d (x ; ) L d (x; ) (1.6) for all x 2 N dn (x ) and all ; 0 2 R m. Note that although we use similar terminologies as in continuous space, SP dn is dierent from SP cn (saddle point in continuous space) ecause they are dened using dierent neighorhoods. The concept of SP dn is very important in discrete prolems ecause, starting from them, we can derive rst-order necessary and sucient condition for CLM dn that leads to gloal minimization procedures. This is stated formally in the following theorem [14]: Theorem 1 First-order necessary and sucient condition on CLM dn. [14] A point in the discrete search space of (1.4) is a CLM dn i it satises (1.6) for any. Theorem 1 is stronger than its continuous counterparts. The rst-order necessary conditions in continuous Lagrange-multiplier theory [9] require CLM cn to e regular points and functions to e dierentiale. In contrast, there are no such requirements for CLM dn. Further, the rst-order conditions in continuous theory [9] are only necessary, and second-order sucient condition must e checked in order to ensure that a point is actually a CLM cn (CLM in continuous space). In contrast, the condition in Theorem 1 is necessary as well as sucient.

5 Constrained Genetic Algorithms and their Applications in Nonlinear Constrained Optimization procedure CSA (; N ) 2. set initial x [x 1; ; x n; 1; ; k ] T with random x, 0; 3. while stopping condition is not satised do 4. generate x 0 2 N dn (x) using G(x; x 0 ); 5. accept x 0 with proaility A T (x; x 0 ) 6. reduce temperature y T T ; 7. end while 8. end procedure a) CSA called with schedule N and rate 1. procedure CSA ID 2. set initial cooling rate 0 and N N 0 ; 3. set K numer of CSA runs at xed ; 4. repeat 5. for i 1 to K do call CSA(; N ); end for; 6. increase cooling schedule N N ; 7. until feasile solution has een found and no etter solution in two successive increases of N ; 8. end procedure ) CSA ID: CSA with iterative deepening Figure 1.2. Constrained simulated annealing algorithm (CSA) and its iterative-deepening extension 2.2. Existing algorithms for nding SPdn Since there is a one-to-one correspondence etween CGM dn and SP dn, it implies that a strategy looking for SP dn with the minimum ojective value will result in CGM dn. We review two methods to look for SP dn. The rst algorithm is the discrete Lagrangian method (DLM) [15]. It is an iterative local search that looks for SP dn y updating the variales in x in order to perform descents of L d in the x suspace, while occasionally updating the variales of unsatised constraints in order to perform ascents in the suspace and to force the violated constraints into satisfaction. When no new proes can e generated in oth the x and suspaces, the algorithm has additional mechanisms to escape from such local traps. It can e shown that the point where DLM stops is a CLM dn when the numer of neighorhood points is small enough to e enumerated in each descent in the x suspace [14, 16]. However, when the numer of neighorhood points is very large and hill-climing is used to nd the rst point with an improved L d in each descent, then the point where DLM stops may e a feasile point ut not necessarily a SP dn. The second algorithm is the constrained simulated annealing (CSA) [13] algorithm shown in Figure 1.2a. It looks for SP dn y proailistic descents in the x suspace and y proailistic ascents in the suspace, with an acceptance proaility governed y the Metropolis proaility. Similar to DLM, if the neighorhood of every point is very large and cannot e enumerated, then the point where CSA stops may only e a feasile point ut not necessarily a SP dn. Using G(x; x 0 ) for generating trial point x 0 in N dn (x), A T (x; x 0 ) as the Metropolis acceptance proaility, and a logarithmic cooling schedule, CSA has een proven to have asymptotic convergence with proaility one to CGM dn [13]. This property is stated in the following theorem: Theorem 2 Asymptotic convergence of CSA. [13] The Markov chain modeling CSA converges to a CGM dn with proaility one. Theorem 2 extends a similar theorem for SA that proves its asymptotic convergence to unconstrained gloal minima of unconstrained optimization prolems. By looking for SP dn in the Lagrangian-function space, Theorem 2 shows the asymptotic convergence of CSA to CGM dn in constrained optimization prolems. Theorem 2 implies that CSA is not a practical algorithm when used to nd CGM dn in one run with certainty ecause CSA will take innite time. In practice, when CSA is run once using a a nite cooling schedule N, it nds a CGM dn with reachaility proaility P R (N ) < 1. To increase its success proaility, CSA with a nite N can e run multiple times from random starting points. Assuming that all the runs are independent, a CGM dn can e found in nite average time dened y (1.3).

6 6 N P R (N ;Q) fail fail fail fail success N opt t 2t 4t 8t 16t Total time for iterative deepening = 31t Optimal time = 12t log 2 (N ) Figure 1.3. An application of iterative deepening in CSA ID. We have veried experimentally that the expected time dened in (1.3) has an asolute minimum at N opt. (Figure 1.1 illustrates the existence of N opt for CSAGA.) It follows that, in order to minimize (1.3), CSA should e run multiple times from random starting points using schedule N opt. To nd N opt at run time without using prolem-dependent information, we have proposed to use iterative deepening [7] y starting CSA with a short schedule and y douling the schedule each time the current run fails to nd a CGM dn [12]. Since the total overhead in iterative deepening is dominated y that of the last run, CSA ID (CSA with iterative deepening in Figure 1.2) has a completion time of the same order of magnitude as that using N opt when the last schedule that CSA is run is close to N opt and that this run succeeds. Figure 1.3 illustrates that the total time incurred y CSA ID is of the same order as the expected overhead at N opt. Note that P R (N opt ) < 1 for one run of CSA at N opt. When CSA is run with a schedule close to N opt and fails to nd a solution, its cooling schedule will e douled and overshoots eyond N opt. To reduce the chance of overshooting into exceedingly long cooling schedules and to increase the success proaility efore its schedule reaches N opt, we have proposed to run CSA multiple times from random starting points at each schedule in CSA ID. Figure 1.2 shows CSA that is run K = 3 times at each schedule efore the schedule is douled. Our results show that such a strategy generally requires twice the average completion time with respect to multiple runs of CSA using N opt efore it nds a CGM dn [12] Genetic algorithms for solving constrained NLP prolems Genetic algorithm (GA) is a general stochastic optimization algorithm that maintains a population of alternative candidates and that proes a search space using genetic operators, such as crossovers and mutations, in order to nd etter candidates. The original GA was developed for solving unconstrained prolems, using a single tness function to rank candidates. Recently, many variants of GA have een developed for solving constrained NLPs. Most of these methods were ased on penalty formulations that use GA to minimize an unconstrained penalty function F(x), consisting of a sum of the ojective and the constraints weighted y penalties. Similar to CSA, these methods do not require the dierentiaility or continuity of functions.

7 Constrained Genetic Algorithms and their Applications in Nonlinear Constrained Optimization 4 7 One penalty formulation is the static-penalty formulation in which all penalties are xed [2]: F (x; ) = f(x) + mx i=1 i jh i (x)j ; (1.7) where > 0, and penalty vector = f 1 ; 2 ; ; m g is xed and chosen to e large enough so that F (x ; ) < F (x; ) 8x 2 X X opt and x 2 X opt : (1.8) Based on (1.8), an unconstrained gloal minimum of (1.7) over x is a CGM dn to (1.4); hence, it suces to minimize (1.7) in solving (1.4). Since oth f(x) and jh i (x)j are lower ounded and x takes nite discrete values, always exists and is nite, therey ensuring the correctness of the approach. Note that other forms of penalty formulations have also een studied in the literature. The major issue of static-penalty methods lies in the diculty of selecting a suitale. If is much larger than necessary, then the terrain will e too rugged to e searched eectively y local-search methods. If it is too small, then feasile solutions to (1.7) may e dicult to nd. Dynamic-penalty methods [6], on the other hand, address the diculties of static-penalty methods y increasing penalties gradually in the following tness function: F(x) = f(x) + (C t) m X j=1 jh i (x)j ; (1.9) where t is the generation numer, and C,, and are constants. In contrast to static-penalty methods, (C t), the penalty on infeasile points, is increased during evolution. Dynamic-penalty methods do not always guarantee convergence to CLM dn or CGM dn. For example, consider a prolem with two constraints h 1 (x) = 0 and h 2 (x) = 0. Assuming that a search is stuck at an infeasile point x 0 and that for all x 2 N dn (x 0 ), 0 < jh 1 (x 0 )j < jh 1 (x)j, jh 2 (x 0 )j > jh 2 (x)j > 0, and jh 1 (x 0 )j + jh 2 (x 0 )j < jh 1 (x)j + jh 2 (x)j, then the search can never escape from x 0 no matter how large (C t) grows. One way to ensure the convergence of dynamic-penalty methods is to use a dierent penalty for each constraint, as in Lagrangian formulation (1.5). In the previous example, the search can escape from x 0 after assigning a much larger penalty to h 2 (x 0 ) than that to h 1 (x 0 ). There are many other variants of penalty methods, such as annealing penalties, adaptive penalties [11] and self-adapting weights [4]. In addition, prolem-dependent operators have een studied in the GA community for handling constraints. These include methods ased on preserving feasiility with specialized genetic operators, methods searching along oundaries of feasile regions, methods ased on decoders, repair of infeasile solutions, co-evolutionary methods, and strategic oscillation. However, most methods require domain-specic knowledge or prolem-dependent genetic operators, and have diculties in nding feasile regions or in maintaining feasiility for nonlinear constraints. In general, local minima of penalty functions are only necessary ut not sucient to e constrained local minima of the original constrained optimization prolems, unless the penalties are chosen properly. Hence, nding local minima of a penalty function does not necessarily solve the original constrained optimization prolem. 3. A General Framework to look for SPdn Although there are many methods for solving constrained NLPs, our survey in the last section shows a lack of a general framework that unies these mechanisms. Without such a framework, it is

8 8 Generate random initial candidate with initial Insert candidate(s) into list ased on sorting criterion (annealing or deterministic) Search in suspace Y Generate new candidate(s) in the susuace (proailistic or greedy) x loop N loop start Generate new candidates in the x suspace (genetic, proailistic, or greedy) N Stopping conditions met Update Lagrangian values of all candidates in list (annealing or determinisic) Y stop Figure 1.4. An iterative stochastic procedural framework to look for SP dn. dicult to know whether dierent algorithms are actually variations of each other. In this section we present a framework for solving constrained NLPs that unies SA, GA, and greedy searches. Based on the necessary and sucient condition in Theorem 1, Figure 1.4 depicts a stochastic procedure to look for SP dn. The procedure consists of two loops: the x loop that updates the variales in x in order to perform descents of L d in the x suspace, and the loop that updates the variales of unsatised constraints for any candidate in the list in order to perform ascents in the suspace. The procedure quits when no new proes can e generated in oth the x and suspaces. The procedure will not stop until it nds a feasile point ecause it will generate new proes in the suspace when there are unsatised constraints. Further, if the procedure always nds a descent direction at x y enumerating all points in N dn (x), then the point where the procedure stops must e a feasile local minimum in the x suspace of L d (x; ), or equivalently, a CLM dn. Both DLM and CSA discussed in Section 2.2 t into this framework, each maintaining a list of one candidate at any time. DLM entails greedy searches in the x and suspaces, deterministic insertions into the list of candidates, and deterministic acceptance of candidates until all constraints are satised. On the other hand, CSA generates new proes randomly in one of the x or variales, accepts them ased on the Metropolis proaility if L d increases along the x dimension and decreases along the dimension, and stops updating when all constraints are satised. In this section, we use genetic operators to generate proes and present in Section 3.1 CGA and in Section 3.2 CSAGA. Finally, we propose in Section 3.3 iterative-deepening versions of these algorithms Constrained genetic algorithm (CGA) CGA in Figure 1.5a was developed ased on the general framework in Figure 1.4. Similar to traditional GA, it organizes a search into a numer of generations, each involving a population of candidate points in a search space. However, it searches in the L d space using genetic operators to generate proes in the x suspace, either greedy or proailistic mechanisms to generate proes in the suspace, and deterministic organization of candidates according to their L d values. Lines 2-3 initialize to zero the generation numer t and the vector of Lagrange multipliers. The initial population P(t) can e either randomly generated or user provided. Lines 4 and 12 terminate CGA when the maximum numer of allowed generations is exceeded. Line 5 evaluates in generation t all candidates in P(t) using L d (x; (t)) as the tness function.

9 Constrained Genetic Algorithms and their Applications in Nonlinear Constrained Optimization procedure CGA(P, N g) 2. set generation numer t 0 and (t) 0; 3. initialize population P(t); 4. repeat /* over multiple generations */ 5. evaluate L d (x; (t)) for all candidates in P(t); 6. repeat /* over proes in x suspace */ 7. y GA(select(P(t))); 8. evaluate L d (y; ) and insert into P(t) 9. until sucient proes in x suspace; 10. (t) (t) ch(h; P(t)); /* update */ 11. t t + 1; 12. until (t > N g) 13. end procedure a) CGA called with population size P and numer of generations N g. 1. procedure CGA ID 2. set initial numer of generations N g = N 0; 3. set K = numer of CGA runs at xed N g; 3. repeat /* iterative deepening to nd CGM dn */ 4. for i 1 to K do call CGA(P; N g) end for 5. set N g N g (typically = 2); 6. until N g exceeds maximum allowed or (no etter solution has een found in two successive increases of N g and N g > 5 N 0 and a feasile solution has een found); 7. end procedure ) CGA ID: CGA with iterative deepening Figure 1.5. Constrained GA and its iterative deepening version. Lines 6-9 explore the x suspace y selecting from P(t) candidates to reproduce using genetic operators and y inserting the new candidates generated into P(t) according to their tness values. After a numer of descents in the x suspace (dened y the numer of proes in Line 9 and the decision ox \search in suspace" in Figure 1.4), the algorithm switches to searching in the suspace. Line 10 updates according to the vector of maximum violations H(h(x); P(t)), where the maximum violation of a constraint is evaluated over all the candidates in P(t). That is, H i (h(x); P(t)) = max H(h i (x)); i = 1; ; m; (1.10) x2p(t) where h i (x) is the i th constraint function, H is the non-negative transformation dened in (1.5), and c is a step-wise constant controlling how fast changes. Operator in Figure 1.5a can e implemented in two ways in order to generate a new. A new can e generated proailistically ased a uniform distriution in ( ch 2 ; ch 2 ], or in a greedy fashion ased on a uniform distriution in (0; ch]. In addition, we can accept new proes deterministically y rejecting negative ones, or proailistically using an annealing rule. In all cases, a Lagrange multiplier will not e changed if its corresponding constraint is satised Comined Constrained SA and GA (CSAGA) Based on the general framework in Figure 1.4, we design CSAGA y integrating CSA in Figure 1.2a and CGA in Figure 1.5a into a comined procedure. The new procedure diers from the original CSA in two respects. First, y maintaining multiple candidates in a population, we need to decide how CSA should e applied to the multiple candidates in a population. Our evaluations show that, instead of running CSA corresponding to a candidate from a random starting point, it is est to run CSA sequentially, using the est solution point found in one run as the starting point of the next run. Second, we need to determine the duration of each run of CSA. This is controlled y parameter q that was set to e Ng after experimental evaluations. The new algorithm shown in 6 Figure 1.6 uses oth SA and GA to generate new proes in the x suspace. Line 2 initializes P(0). Unlike CGA, any candidate x = [x 1 ; ; x n ; 1 ; ; k ] T in P(t) is dened in the joint x and suspaces. Initially, x can e user-provided or randomly generated, and is set to zero.

10 10 1. procedure CSAGA(P; N g) 2. set t 0, T 0, 0 < < 1, and P(t); 3. repeat /* over multiple generations */ 4. for i 1 to q do /* SA in Lines 5-10 */ 5. for j 1 to P do 6. generate x 0 j from N dn (x j ) using G(x j ; x 0 j); 7. accept x 0 j with proaility A T (x j ; x 0 j) 8. end for 9. set T T ; /* set T for the SA part */ 10. end for 11. repeat /* y GA over proes in x suspace */ 12. y GA(select(P(t))); 13. evaluate L d (y; ) and insert y into P(t); 14. until sucient numer of proes in x suspace; 15. t t + q; /* update generation numer */ 16. until (t N g) 17. end procedure Figure 1.6. CSAGA: Comined CSA and CGA called with population size P and N g generations. Lines 4-10 perform CSA using q proes on every candidate in the population. In each proe, we generate proailistically x 0 and accept it ased on the Metropolis proaility. Experimentally, j we set q to e Ng. As discussed earlier, we use the est point of one run as the starting point of 6 the next run. Lines start a GA search after the SA part has een completed. The algorithm searches in the x suspace y generating proes using GA operators, sorting all candidates according to their tness values L d after each proe is generated. In ordering candidates, since each candidate has its own vector of Lagrange multipliers, the algorithm rst computes the average value of Lagrange multipliers for each constraint over all candidates in P(t) and then calculates L d for each candidate using the average Lagrange multipliers. Note that CSAGA has diculties similar to those of CGA in determining a proper numer of candidates to use in its population and the duration of each run. We address these two issues in the CSAGA ID in the next susection CGA and CSAGA with iterative deepening In this section we present a method to determine the optimal numer of generations in one run of CGA and CSAGA in order to nd a CGM dn. The method is ased on the use of iterative deepening [7] that determines an upper ound on N g in order to minimize the expected total overhead in (1.3), where N g is the numer of generations in one run of CGA. The numer of proes expended in one run of CGA or CSAGA is N = N g P, where P is the population size. For a xed P, let ^PR (N g ) = P R (P N g ) e the reachaility proaility of nding CGM dn. From (1.3), the expected total numer of proes using multiple runs of either CGA or CSAGA and xed P is: N P R (N) = N gp P R (N g P ) = P N g ^P R (N g ) (1.11) N In order to have an optimal numer of generations N gopt that minimizes (1.11), g must have ^P R (N g) an asolute minimum in (0,1). This condition is true since ^PR (N g ) of CGA has similar ehavior

11 Constrained Genetic Algorithms and their Applications in Nonlinear Constrained Optimization 6 11 as P R (N g ) of CSA. It has een veried ased on statistics collected on ^PR (N g ) and N g at various P when CGA and CSAGA are used to solve ten discretized enchmark prolems G1-G10 [11]. Figure 1.1 illustrates the existence of such an asolute minimum when CSAGA with P = 3 was applied to solve G1. Similar to the design of CSA ID, we apply iterative deepening to estimate N gopt. CGA ID in Figure 1.5 uses a set of geometrically increasing N g to nd a CGM dn : N gi = i N 0 ; i = 0; 1; : : : (1.12) where N 0 is the (small) initial numer of generations. Under each N g, CGA is run for a maximum of K times ut stops immediately when a feasile solution has een found, when no etter solution has een found in two successive generations, and after the numer of iterations has een increased geometrically at least ve times. These conditions are used to ensure that iterative deepening has een applied adequately. For iterative deepening to work, > 1. Let ^PR (N gi ) e the reachaility proaility of one run of CGA under N gi generations, B opt (f 0 ) e the expected total numer of proes taken y CGA with N gopt to nd a CGM dn, and B ID (f 0 ) e the expected total numer of proes taken y CGA ID in Figure 1.5 to nd a solution of quality f 0 starting from N 0 generations. According to (1.11), B opt (f 0 ) = P N gopt ^P R (N gopt ) (1.13) The following theorem shows the sucient conditions in order for B ID (f 0 ) = O(B opt (f 0 )). Theorem 3 Optimality of CGA ID and CSAGA ID. B ID (f 0 ) = O(B opt (f 0 )) if a) ^PR (0) = 0; ^PR (N g ) is monotonically non-decreasing for N g in (0; 1); and lim Ng!1 ^P R (N g ) 1; ) (1 ^PR (N gopt )) K < 1. The proof is not shown due to space limitations. Typically, = 2, and ^PR (N gopt ) 0:25 in all the enchmarks tested. Sustituting these values into condition () in Theorem 3 yields K > 2:4. In our experiments, we have used K = 3. Since CGA is run a maximum of three times under each N g, B opt (f 0 ) is of the same order of magnitude as one run of CGA with N gopt. The only remaining issue left in the design of CGA ID and CSAGA ID is in choosing a suitale population size P in each generation. In designing CGA ID, we found that the optimal P ranges from 4 to 40 and is dicult to determine a priori. Although it is possile to choose a suitale P dynamically, we do not present the algorithm here due to space limitations and ecause it performs worse than CSAGA ID. In selecting P for CSAGA ID, we note in the design of CSA ID in Figure 1.2 that K = 3 parallel runs are made at each cooling schedule in order to increase the success proaility of nding a solution. For this reason, we set P = K = 3 in our experiments. Our experimental results in the next section show that, although the optimal P may e slightly dierent from 3, the corresponding expected overhead to nd a CGM dn diers very little from that when a constant P is used.

12 12 4. Experimental Results We present in this section our experimental results in evaluating CSA ID, CGA ID and CSAGA ID on discrete constrained NLPs. Based on the framework in Figure 1.4, we rst determine the est comination of strategies to use in generating proes and in organizing candidates. Using the est comination of strategies, we then show experimental results on some constrained NLPs. Due to a lack of large-scale discrete enchmarks, we derive our enchmarks from two sets of continuous enchmarks: Prolem G1-G10 [11, 8] and Floudas and Pardalos' Prolems [5] Implementation Details In theory, algorithms derived from the framework, such as CSA, CGA, and CSAGA, will look for SP dn. In practice, however, it is important to choose appropriate neighorhoods and generate proper trial points in x and suspaces in order to solve constrained NLPs eciently. An important component of these methods is the frequency at which is updated. Like in CSA [13], we have set experimentally in CGA and CSAGA the ratio of generating trial points in x and suspaces from the current point to e 20n to m, where n is the numer of variales and m is the numer of constraints. This ratio means that x is updated more often than. In generating trial points in the x suspace, we have used a dynamically controlled neighorhood size in the SA part [13] ased on the 1:1 ratio rule [3], whereas in the GA part, we have used the seven operators in Genocop III [10] and L d as our tness function. In implementing CSA ID, CGA ID and CSAGA ID, we have used the default parameters of CSA [13] in the SA part and those of Genocop III [10] in the GA part. The generation of trial point 0 in the suspace is done y the following rule: 0 j = j + r 1 j where j = 1; ; m: (1.14) Here, r 1 is randomly generated in [1=2; +1=2] if we choose to generate proailistically, and is randomly generated in [0; 1] if we choose to generate proes in in a greedy fashion. We adjust adaptively according to the degree of constraint violations, where = w H(x) = [w 1 H 1 (x); w 2 H 2 (x); ; w m H m (x)]; (1.15) represents vector product, and H is the vector of maximum violations dened in (1.10). When H i (x) is satised, i does not need to e updated; hence, i = 0. In contrast, when a constraint is not satised, we adjust i y modifying w i according to how fast H i (x) is changing: 0 w w i = i 1 w i if H i (x) > 0 T if H i (x) < 1 T (1.16) where T is the temperature, and 0 = 1:25, 1 =0.8, 0 = 1:0, and 1 = 0:01 were chosen experimentally. When H i (x) is reduced too quickly (i.e., H i (x) < 1 T ), H i (x) is over-weighted, leading to possily poor ojective values or diculty in satisfying other under-weighted constraints. Hence, we reduce i 's neighorhood. In contrast, if H i (x) is reduced too slowly (i.e., H i (x) > 0 T ), we enlarge i 's neighorhood in order to improve its chance of satisfaction. Note that w i is adjusted using T as a reference ecause constraint violations are expected to decrease when T decreases. In addition, for iterative deepening to work, we have set the following parameters: = 2, K = 3, N 0 = 10 n v, and N max = 1: n v, where n v is the numer of variales, and N 0 and N max are, respectively, initial and maximum numer of proes.

13 Constrained Genetic Algorithms and their Applications in Nonlinear Constrained Optimization 7 13 Tale 1.1. Timing results on evaluating various cominations of strategies in CSA ID, CGA ID and CSAGA ID with P = 3 to nd solutions that deviate y 1% and 10% from the est-known solution of a discretized version of G2. All CPU times in seconds were averaged over 10 runs and were collected on a Pentinum III 500-MHz computer with Solaris 7. '' means that no solution with desired quality can e found. Proe Generation Strategy Insertion Target Solution 1% o CGM dn Target Solution 10% o CGM dn suspace x suspace Strategy CSA ID CGA ID CSAGA ID CSA ID CGA ID CSAGA ID proailistic proailistic annealing 6.91 sec sec sec sec sec. proailistic proailistic deterministic 9.02 sec sec sec sec sec. proailistic deterministic annealing sec sec sec. proailistic deterministic deterministic sec sec. greedy proailistic annealing 7.02 sec sec sec sec. greedy proailistic deterministic 7.02 sec sec sec sec. greedy deterministic annealing sec sec sec. greedy deterministic deterministic sec sec sec Evaluation Results Due to a lack of large-scale discrete enchmarks, we derive our enchmarks from two sets of continuous enchmarks: Prolem G1-G10 [11, 8] and Floudas and Pardalos' Prolems [5]. In generating a discrete constrained NLP, we discretize continuous variales in the original continuous constrained NLP into discrete variales as follows. In discretizing continuous variale x i in range [l i ; u i ], where l i and u i are lower and upper ounds of x i, respectively, we force x i to take values from the set: A i = 8 < : n o a i + ia i j; j = 0; 1; ; s if s i a i < 1 ai + 1 s j; j = 0; 1; ; ( i a i )sc if i a i 1; (1.17) where s = 1: Tale 1.1 shows the results of evaluating various cominations of strategies in CSA ID, CGA ID, and CSAGA ID on a discretized version of G2 [11, 8]. We show the average time of 10 runs for each comination in order to reach two solution quality levels (1% or 10% worse than CGM dn, assuming the value of CGM dn is known). Evaluation results on other enchmark prolems are similar and are not shown due to space limitations. Our results show that CGA ID is usually less ecient than CSA ID or CSAGA ID. Further, CSA ID or CSAGA ID has etter performance when proes generated in the x suspace are accepted y annealing rather than y deterministic rules (the former prevents a search from getting stuck in local minima or infeasile points). On the other hand, there is little dierence in performance when new proes generated in the suspace are accepted y proailistic or y greedy rules and when new candidates are inserted according to annealing or deterministic rules. In short, generating proes in the x and suspaces proailistically and inserting candidates in oth the x and suspaces y annealing rules leads to good and stale performance. For this reason, we use this comination of strategies in our experiments. We next test our algorithms on ten constrained NLPs G1-G10 [11, 8]. These prolems have ojective functions of various types (linear, quadratic, cuic, polynomial, and nonlinear) and constraints of linear inequalities, nonlinear equalities, and nonlinear inequalities. The numer of variales is up to 20, and that of constraints, including simple ounds, is up to 42. The ratio of feasile space with respect to the whole search space varies from 0% to almost 100%, and the topologies of feasile regions are quite dierent. These prolems were originally designed to e solved y evolutionary

14 14 Tale 1.2. Results on CSA ID, CGA ID and CSAGA ID in nding the est-known solution f for 10 discretized constrained NLPs and their corresponding results found y EA. (S.T. stands for strategic oscillation, H.M. for homomorphous mappings, and D.P. for dynamic penalty. B ID(f ), the CPU time in seconds to nd the est-known solution f, were averaged over 10 runs and were collected on a Pentinum III 500-MHz computer with Solaris 7. #L d represents the numer of L d (x; )-function evaluations. The est B ID(f ) for each prolem is oxed.) Prolem Best EAs CSA ID CGA ID CSAGA ID ID Solution f Best Found Method B ID(f ) #L d P opt B ID(f ) P B ID(f ) #L d P opt B ID(f ) G1 (min) Genocop 1.65 sec sec sec G2 (max) S.T sec sec. 3 G3 (max) S.T sec sec. 3 G4 (min) H.M sec sec sec sec sec sec sec sec sec. G5 (min) D.P sec sec sec G6 (min) Genocop 0.99 sec sec sec G7 (min) H.M sec sec sec G8 (max) H.M sec sec sec G9 (min) Genocop 0.74 sec sec. 3 G10 (min) H.M sec sec sec sec sec sec sec sec sec sec. algorithms (EAs) in which constraint handling techniques were tuned for each prolem in order to get good results. Examples of such techniques include keeping a search within feasile regions with specic genetic operators and dynamic and adaptive penalty methods. Tale 1.2 compares the performance of CSA ID, CGA ID, and CSAGA ID with respect to B ID (f ), the expected total CPU time of multiple runs until a solution of value f is found. The rst two columns show the prolem IDs and the corresponding known f. The next two columns show the est solutions otained y EAs and the specic constraint handling techniques used to generate the solutions. Since all CSA ID, CGA ID and CSAGA ID can nd a CGM dn in all 10 runs, we compare their performance with respect to T, the average total overhead of multiple runs until a CGM dn is found. The fth and sixth columns show, respectively, the average time and numer of L d (x; ) function evaluations CSA ID takes to nd f. The next two columns show the performance of CGA ID with respect to P opt, the optimal population size found y enumeration, and the average time to nd f. These results show that CGA ID is not competitive as compared to CSA ID, even when P opt is used. The results on including additional steps in CGA ID to select a suitale P at run time are worse and are not shown due to space limitations. Finally, the last ve columns show the performance of CSAGA ID. The rst three present the average times and numer of L d (x; ) evaluations using a constant P, whereas the last two show the average times using P opt found y enumeration. These results show little improvements in using P opt. Further, CSAGA ID has etween 9% and 38% in improvement in B ID (f ), when compared to that of CSA ID, for the 10 prolems except for G4 and G10. Comparing CGA ID and CSAGA ID with EA, we see that EA was only ale to nd f in three of the ten prolems, despite extensive tuning and using prolem-specic heuristics, whereas oth CGA ID and CSAGA ID can nd f for all these prolems without any prolem-dependent strategies. It is not possile to report the timing results of EA ecause the results are the est among many runs after extensive tuning. Finally, Tale 1.3 shows the results on selected discretized Floudas and Pardalos' enchmarks [5] that have more than 10 variales and that have many equality or inequality constraints. The rst three columns show the prolem IDs, the known f, and the numer of variales (n v ) in each. The

15 Constrained Genetic Algorithms and their Applications in Nonlinear Constrained Optimization 8 15 Tale 1.3. Results on CSA ID and CSAGA ID with P = 3 in solving selected Floudas and Pardalos' discretized constrained NLP enchmarks (with more than n v = 10 variales). Since Prolem 5: and 7: are especially large and dicult and a search can rarely reach their true CGM dn, we consider a CGM dn found when the solution quality is within 10% of the true CGM dn. All CPU times in seconds were averaged over 10 runs and were collected on a Pentium-III 500-MHz computer with Solaris 7. Prolem f(x) CSA ID CSAGA ID ID Best f n v B ID(f ) B ID(f ) 2.7.1(min) sec (min) sec (min) sec (min) sec (min) sec. 5.2(min) sec. 5.4(min) sec. 7.2(min) sec. 7.3(min) sec. 7.4(min) sec sec sec sec sec sec sec sec sec sec sec. last two columns compare B ID (f ) of CSA ID and CSAGA ID with xed P = 3. They show that CSAGA ID is consistently faster than CSA ID (etween 1.3 and 26.3 times), especially for large prolems. This is attriuted to the fact that GA maintains more diversity of candidates y keeping a population, therey allowing competition among the candidates and leading SA to explore more promising regions. 5. Conclusions In this chapter we have presented new algorithms to look for discrete-neighorhood saddle points in discrete Lagrangian space of constrained optimization prolems. Our results show that genetic algorithms, when comined with simulated annealing, are eective in locating saddle points. Future developments will focus on etter ways to select appropriate heuristics in proe generation, including search direction control and neighorhood size control, at run time.

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