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Similarly, plots (c and d) of figure 3.2 show the results for experiment 2 where BTS is utilized. From the crossover plot (c) it is interesting to note how the envelope for the [MATH] shrinks towards zero as the curve for the change in the average fitness settles around zero on the fitness axis. This is indicative of t... |
On the other hand, an observation of plot 3.2 (d) reveals that the curve for the change in the average fitness due to mutation operator lie and remain beneath zero on the fitness axis throughout the evolution. Also, larger portion of the shaded area for the [MATH] envelope is beneath zero . This is contrary to its beha... |
3.5.1 The proposed convergence threshold parameter Having used one standard deviation interval to analyse the effect of genetic operators, then, the width of the [MATH] envelope for the effect of an operator [MATH] (crossover in this case) on the fitness growth at every [MATH] th generation lies within the interval: |
[EQUATION] where [MATH] is the change in the average fitness of the population at iteration [MATH] due to operator [MATH] and [MATH] is the corresponding standard deviation. Let the width be represented by [MATH] , then it can be determined as follows: |
[EQUATION] Consider the complete set of plots for experiment 1 (for BTS) shown on figure 3.3 , the point labelled A on plot (a) directly corresponds to the point labelled A on plot (c) when read from the x-axis (i.e., the generations axis). It is fairly easy to notice that the more the width of the [MATH] envelope for ... |
Therefore, the proposed convergence measure is to prescribe a threshold value for the parameter [MATH] such that whenever [MATH] falls below this threshold, the evolutionary search process is automatically terminated. We must remark here that this threshold parameter is user defined and its appropriate value is determi... |
3.6 Contribution This chapter provides insight on the relevance of convergence detection in EC and sheds light on some conventional convergence measures. Most importantly, it provides insight on the interactions among the genetic operators during the evolutionary search process. A visual means for investigating the ind... |
The experimental results obtained after comparing the plots for the contribution of the genetic operators using RWS and BTS reveal two interesting findings. |
First, while using the RWS method, the proposed technique for visualizing the effect of genetic operators using extended Price’s equation have shown that all the three operators play random roles of exploiting and exploring the search space throughout the evolution period. This has made it infeasible to deploy their in... |
Second, substituting the RWS with a BTS method reveals the contrary. The extended Price’s plots in this case demonstrate a clear separation of roles among the three genetic operators. The BTS operator takes the lead in exploiting the highly fit areas of the search space; the mutation operator handles the exploration as... |
Consequently, the insight gathered above led to the development of a novel convergence measure (section 3.5 ) that can allow automatic convergence detection in EC. |
3.7 Remarks While the earlier chapters have introduced and analysed the initialization and developmental aspects of evolutionary computation algorithms, this chapter augments the previous work by analysing the process that could lead to successful termination of the evolutionary search, which is a prerequisite to achie... |
Chapter 4 Local Search Algorithms for Optimization In the previous chapters, detailed investigation and analysis on evolutionary computations was given. Such algorithms are global search approximate techniques where success relies upon some stochastic heuristics. The design of the proposed hybrid algorithm requires com... |
4.1 An Overview of Local Optimization Algorithms Consider a general expression of a continuous optimization problem shown in equation ( 4.1 ), the objective function to be optimized |
[MATH] is defined in terms of a vector of the design variables [MATH] of length [MATH] that is in the set of real numbers [MATH] |
[EQUATION] Commonly, gradient based optimization algorithms sequentially generate at every [MATH] iteration, a vector of solution points [MATH] that is expected to terminate at [MATH] (i.e the optimal solution) when either no more progress can be made or when the optimal solution has been attained with sufficient accur... |
Typically, every new solution point [MATH] is expected to yield a lower function value than its predecessor [MATH] . In fact, a critical distinction between the various local optimization algorithms is on the nature of their successive iterations. A class of algorithms that insists on reduction in the function value at... |
. The Newton and quasi-Newton algorithms are part of this class. The other class that do not insist on minimizing the value of the objective function at the end of every iteration usually enforces [MATH] where [MATH] is the maximum acceptable iterations without a decrease in the objective function value. This is the cl... |
Another major categorization of the gradient based algorithms is on their approach for stepping from one iteration point [MATH] to the next [MATH] . The approach for taking a step is always either a line search or a trust region based. The general expression for deriving the next iteration point [MATH] is dependent on ... |
[EQUATION] The line search based algorithms evaluate the search direction and then decide on how long to search along that direction by estimating a suitable value for the step length parameter [MATH] . On the contrary, trust region based algorithms start by defining a region around the current iteration point [MATH] w... |
One might have noticed that for the line search based methods, if the step length parameter does not lead to a decrease in the value of the objective function, the algorithm can easily try to re-evaluate a feasible one. However, in trust region methods, both the search direction and the step length must be discarded, t... |
have the view that trust region methods can be more reliable compared to the line search based methods especially when the initial starting point is significantly away from the actual minimizer. Yet, since the local search algorithm (SQP) to be proposed in the later sections will be a line search based method, this inv... |
4.2 Line Search Based Local Optimization Methods As highlighted earlier, the iterative progress in line search based methods relied on the computed search direction [MATH] and the evaluation of the possible distance along the direction to which the search can progress, i.e. step length [MATH] . The manner in which the ... |
For any smooth continuously differentiable function [MATH] in the neighbourhood of iteration point [MATH] , and an assumed vector of search direction [MATH] , the necessary and sufficient conditions for optimality requires [MATH] , and [MATH] to be a symmetric positive definite matrix for [MATH] to be a local minimizer... |
[EQUATION] Gradient based methods assume that the objective function [MATH] is differentiable and it is approximately quadratic (i.e. convex) in the vicinity of the stationary point [MATH] . Therefore, the second-order Taylor expansion of equation ( 4.3 ) is: |
[EQUATION] Thus, setting the gradient of equation ( 4.4 ) to zero at stationary point and solving for [MATH] yields: [EQUATION] where [MATH] is the Hessian of the function [MATH] and [MATH] is the search direction which is required to be a descent direction. |
4.3 Search Directions and Step Length in Gradient Based Algorithms As described previously, any step of the gradient based algorithms involves evaluation of the search direction [MATH] and the step length parameter [MATH] such that the next iteration is defined as in equation ( 4.2 ) above. In the following, the nature... |
4.3.1 Methods of Evaluating Descent Search Directions Commonly, gradient based algorithms require the search direction [MATH] at any given iteration [MATH] to be a descent direction such that the directional derivative: |
[EQUATION] will guarantee a reduction in the value of the objective function [MATH] along this direction. From equation ( 4.5 ), a general form for the search directions [MATH] is: |
[EQUATION] where [MATH] is a symmetric non-singular matrix. The most commonly used types of search directions for local optimization algorithms are as follows: |
i. Steepest Descent Direction: The search direction [MATH] is called the steepest descent direction and among all the directions via which the search could move from point [MATH] to [MATH] , this is the direction along which [MATH] decreases most rapidly. Therefore, steepest descent algorithms are line search methods t... |
ii. Newton Direction: Another important search direction is the Newton direction . Derived from the second-order Taylor expansion of [MATH] , the value of [MATH] is the true Hessian of the objective function. The Newton direction is quite a reliable descent direction if the Hessian [MATH] is sufficiently smooth so that... |
The main drawback of Newton methods is the need for the evaluation of the true Hessian [MATH] which is a matrix of second derivatives and can be quite cumbersome, error prone and expensive especially when the dimension of [MATH] is large. Furthermore, whenever [MATH] is not positive definite, the Newton direction may n... |
iii. Quasi-Newton Direction: Yet another search direction is the quasi-Newton direction . In this case, instead of evaluating the true Hessian [MATH] , the value of [MATH] 4.7 ) is only a mere approximation of it. The initial approximation of the Hessian [MATH] is usually an identity matrix [MATH] , which is then updat... |
A common drawback of quasi-Newton methods is that their rate of convergence is slower than that of Newton methods as they require running through several iterations. Also, after certain number of steps, the Hessian approximation often tends to yield an ill-conditioned matrix that may cause the entire search process to ... |
, the initial approximation of [MATH] is an identity matrix [MATH] , the entire search process is always reduced to a simple steepest descent after every reset of the Hessian matrix. |
iv. Conjugate Gradient Direction: This category of search directions are typically more effective than steepest descent directions and are fairly computationally equivalent. Unlike in the Newton and quasi-Newton directions, this search direction requires no storage of large Hessian matrices. The methods that use this s... |
, conjugate gradient methods may fail to converge on non-convex problems and therefore need restart. 4.3.2 Methods of Evaluating the Step Length Parameter |
The step length parameter [MATH] is a positive scalar that is expected to give a substantial reduction in the function value. It is normally the minimizer of the merit function defined by: |
[EQUATION] Generally, the ideal value for the step length parameter [MATH] is the global minimizer of the merit function ( 4.8 ) itself. Determination of this global minimizer requires several evaluations of the objective function [MATH] and its gradient [MATH] . Therefore, since evaluation of even a local minimizer of... |
[EQUATION] Worth noting is that the step length [MATH] need not to lie near the minimizer of the merit function [MATH] for it to effectively yield a sufficient reduction in the objective function. |
Several inexact line search algorithms for estimating a suitable value for the step length parameter exist, and they are usually named after the termination condition used. Thus, they include the method based on Armijo’s rule, Goldstein condition and the popular Wolf conditions |
. All these algorithms typically possess two key stages. The first stage is a backtracking procedure that finds an interval containing the desirable step lengths, while the second is an interpolation phase where a good value for [MATH] is computed within the obtained interval. |
The most commonly used termination condition is based on the two Wolf conditions and it has been analytically proven that , to every smooth continuous function there exist a value for the step length parameter that satisfies the following two Wolf conditions. |
First: The step length [MATH] must provide sufficient decrease in [MATH] such that: [EQUATION] where [MATH] is a positive constant which is quite small in practice, typically [MATH] . Thus, the reduction in [MATH] should be proportional to both the value of [MATH] and the directional derivative [MATH] |
Second: This is called the curvature condition, it ensures that the derivative of the merit function [MATH] (see equation ( 4.8 )) is greater than a constant [MATH] times the derivative of [MATH] , i.e.: |
[EQUATION] such that the new directional derivative satisfies: [EQUATION] where [MATH] , its typical value for Newton and quasi-Newton methods is [MATH] and [MATH] for nonlinear conjugate gradient methods. For more details on these and other techniques for evaluating the step length parameter, see |
4.4 Convergence Analysis of Gradient Based Algorithms Convergence assessment is critical to evaluating the performance of any optimization algorithm. In this section, a brief investigation on the rate of convergence and global convergence of the gradient based algorithms will be presented and the local search algorithm... |
While a number of measures exist for evaluating the rate of convergence of gradient based algorithms, the commonly used one is the [MATH] convergence ( [MATH] stands for quotient) measure defined in terms of the quotient of successive errors. For a sequence of solution points [MATH] that iteratively converges to a mini... |
[MATH] Linear: The convergence is [MATH] linear if there exist a constant [MATH] such that: [EQUATION] [MATH] Superlinear: An algorithm is said to converge [MATH] superlinearly if as the number of iterations [MATH] tends to infinity, the error between two successive iteration points dies out. |
[EQUATION] [MATH] Quadratic: These are algorithms which for any scalar [MATH] , their rate of convergence satisfies: [EQUATION] It is easy to notice that higher order rate of convergence are also possible and the trend is that a [MATH] quadratic algorithm will always converge faster than a [MATH] superlinear or [MATH] ... |
Conversely, the global convergence of gradient based algorithms requires not only a suitable estimate of the step length parameter, but also a carefully chosen search direction. In order to have good convergence characteristics, the evaluated search direction must not be orthogonal to the gradient [MATH] , i.e. at leas... |
From the foregoing, we would like to remark that one undesirable behaviour associated with general gradient based algorithms is that while on one hand steepest descent algorithms have guaranteed global convergence, their rate of convergence is quite slow (i.e., converge only Q-linearly). On the other hand, Newton algor... |
A summary of the key features for the aforementioned major categories of gradient based algorithms is given in table 4.1 . It provides a simple comparison of these methods from which it can be deduced that Newton based optimization algorithm is the most suitable choice for the proposed hybrid algorithm. This is because... |
4.5 Sequential Quadratic Programming Algorithm The standard sequential quadratic programming (SQP) algorithm is a Newton based nonlinear optimization algorithm that can be implemented either in the line search or trust region framework and can successfully handle constrained nonlinear problems. One major feature that i... |
[EQUATION] The framework of SQP is essentially based on Newton method, and since Newton methods solve nonlinear problems via a sequence of Newton steps, this method is called sequential quadratic programming. Extensive treatment on SQP algorithm can be found in the books of Fletcher and Nocedal |
While the standard SQP algorithm uses active set strategy that is based on the null-space and range-space methods to solve its QP subproblems, the method adopted here is based on the earlier work in |
where interior point method (see section 4.6 ) will be deployed to minimize the QP problem. 4.6 Interior Point Method for Solving Quadratic |
Problems Interior point methods (IPM) have their origin from linear programming (LP) and have become important tools in mathematical programming, operations research and in many other areas of science. The main idea in IPM algorithms is to approach the optimal solution of the LP problem through the interior of the feas... |
which moves along the boundary of the feasible region. IPM approach was also proposed by Hoppe Over the last two decades, a lot of research works have been reported on IPM algorithm and its variants many of which are surveyed and referenced by Mizuno et al. |
and Wright . Interior point methods can also be extended to handle general convex problems such as quadratic programming problems. Details can be found in the book of Nesterov and Nemirovskii |
and in Hertog In this work, IPM algorithm will be employed to minimize the QP subproblems produced as a result of linearization of the original nonlinear optimization problem by the main SQP algorithm. Thus, IPM will form the inner (optimization) loop of the proposed SQP algorithm. |
Let a general quadratic programming problem be defined as: [EQUATION] then, the function [MATH] 4.17 ) is convex if at least the matrix of quadratic terms [MATH] is symmetric positive semidefinite. |
If the function to be optimized is subject to some constraints [MATH] , say [EQUATION] then, IPM algorithms use the method of Lagrange multipliers to redefine the optimization problem into a composite function [MATH] called Lagrange function (named after its inventor Joseph Louis Lagrange |
). The Lagrange function combines the objective function ( 4.17 ) and all the constraints ( 4.18 ) such that: [EQUATION] where [MATH] is a vector of Lagrange multipliers. |
At every iteration of the SQP algorithm, the matrix of quadratic terms [MATH] is derived from the Hessian of the original nonlinear problem. [MATH] is then used to construct the QP subproblem ( 4.17 ) to be solved in the inner loop of the SQP algorithm. The minimum of ( 4.19 ) can then be obtained by taking its gradien... |
is the reason why the method of Lagrange multipliers can be quite handy. Hence, minimizing the Lagrange function yields the minimum of the originally constrained function. Details on constrained minimization can be found in |
Of particular interest at this juncture is the fact that standard implementations of SQP algorithm only utilize approximate Hessians for the nonlinear problems. The Hessian is initially approximated by an identity matrix [MATH] and then updated iteratively via the well known Hessian update procedure proposed by Broyden... |
The proposal here is to develop an automatic differentiation package that can algorithmically determine the exact values for the gradient and Hessian of any differentiable function at any given search point [MATH] . The cost of evaluating the derivatives will be more or less equivalent to that of evaluating the functio... |
i. Having exact Hessians at its disposal, the proposed SQP algorithm has upgraded the standard SQP algorithm from a mere quasi-Newton algorithm that takes steepest descent steps to a Newton algorithm taking full Newton steps at every iteration. Hence, the convergence rate is enhanced from superlinear to quadratic. |
ii. Since exact Hessian matrices will be used, the frequent reset of the search direction due to ill-conditioned approximate Hessians could be avoided or at least minimized . This means both global convergence and quadratic rate of convergence are assured. Therefore, the proposed method is hoped to converge to the mini... |
Algorithm shows a detailed framework for the proposed SQP local optimization algorithm. The while loop at line [MATH] exits only if one of the three stopping conditions is satisfied. The first two are tolerance parameters [MATH] and [MATH] which are small positive numbers used to check whether the delta changes in the ... |
Algorithm 1 The SQP local optimization algorithm 1: begin 2: [MATH] 3: [MATH] [MATH] is the starting point returned by the EC algorithm |
4: [MATH] % Initial search direction [MATH] is a vector of all ones 5: while [MATH] and [MATH] and [MATH] do 6: Linearize ( 4.1 ) into a QP subproblem ( 4.17 |
7: [MATH] % computed via Automatic Differentiation 8: Evaluate [MATH] by minimizing QP subproblem ( 4.17 ) % IPM algorithm will be used |
9: if [MATH] satisfies Wolf conditions ( 4.10 ) and ( 4.11 then 10: [MATH] 11: else 12: Evaluate [MATH] that satisfies ( 4.10 ) and ( 4.11 |
13: end if 14: [MATH] 15: [MATH] 16: end while 17: end In the following, details of the exact Hessian evaluation using automatic differentiation principle will be given. |
4.7 Automatic Differentiation for Exact Derivatives Evaluation Also called algorithmic differentiation, automatic differentiation (AD) is built on the notion that the value of any function, simple or complex, is evaluated via a sequence of basic elementary operations involving unary or binary operators and their operan... |
Unlike the finite differencing method which has its origin from Taylor’s theorem, AD is based on repeated application of the chain rule from elementary calculus. Recall that according to chain rule, if [MATH] is a composite function defined in terms of a vector [MATH] which is in turn a function of [MATH] , then, the d... |
[EQUATION] Hence, AD techniques apply chain rule on a computational representation (tree-like graphs) of a function to generate analytic values for the function and its derivatives. |
Historically, the basic ideas of AD have been around for long time . However, it was the early extensive study by Griewank et al. |
that revived the interest in the use of algorithmic methods for evaluation of derivatives. Thereafter, a number of researches have been published on AD principles and its applications in mathematics and machine learning by Christianson and colleagues |
and recently in 4.7.1 Graphical Representation and Algorithmic Evaluation of Functions Consider the two dimensional function shown in equation ( 4.21 ), evaluating the function values at any given point [MATH] will entail an orderly execution of the elementary operations that made up the function. |
[EQUATION] The elementary operations will lead to defining some intermediate variables that link the input variables to the output. Note that all the input, intermediate and the output variables can be represented as vertices of a tree that graphically represents the entire function. The list in table 4.2 gives the det... |
Consider the tree-like graph shown in figure 4.1 , it gives a graphical representation of problem ( 4.21 ). The root vertices represent the input variables. The intermediate vertices stand for the intermediate variables and the top vertex is the output variable. According to graph theory, the ordering require that for ... |
Therefore, the value of the child vertex [MATH] must be evaluated before the parent vertex [MATH] , i.e., the values of all the children vertices must be obtained prior to evaluating that of their parent. Hence, the overall function value can be obtained by evaluating the vertices in the graph from root through the top... |
It is worth noting that when it comes to implementing the AD tool (see section 4.7.3 ), the user need not to explicitly break down the function into its elementary components as in the listing of table 4.2 , but the identification of the intermediate variables and the graphical structure is usually done by the compiler... |
4.7.2 Modes of Automatic Differentiation and their Complexities The two basic modes via which AD can algorithmically yield the values and derivatives of any differentiable function are the forward and reverse modes. Although forward mode will be adopted here, the reverse mode will also briefly be presented. |
Forward mode of AD This is also called forward accumulation, it derives its name from the fact that the derivative evaluation sweeps in the same direction as that of evaluating the function value. First, the computer interprets the function (such as problem ( 4.21 )) as a sequence of elementary operations on the work v... |
Therefore, for a function of [MATH] dimensions , evaluating its complete gradient will require [MATH] forward sweeps; in our example problem ( 4.21 ), the function has 2 independent variables and will therefore need 2 forward sweeps to get the derivatives with respect to the 2 variables. It is easy to realize that forw... |
[EQUATION] The listing in table 4.3 shows the expressions for the forward derivatives of all the vertices propagated from the input, intermediate to the output vertex. At the beginning of every forward sweep, the derivative of one of the input vertices is seeded to [MATH] and all other inputs to zero. In other words, w... |
The above principle explains the basic forward accumulation method of evaluating derivatives. Notice from the listing in table 4.3 that if the value of [MATH] is seeded to [MATH] and [MATH] to [MATH] , then, the value of [MATH] will be the derivative of the function with respect to [MATH] . Conversely, if [MATH] is see... |
A more efficient and elegant approach that will be implemented in this work, defines the initial seeds for the vertices of the independent variables as vectors rather than scalars, i.e., the vertices are set to [MATH] and [MATH] . Notice that the two vectors [MATH] and [MATH] now correspond to the rows of an identity m... |
Reverse mode of AD Also called reverse accumulation, this mode acquired its name from the fact that instead of evaluating the derivative of every intermediate variable (from the root to top of figure 4.1 ) with respect to a chosen input variable [MATH] , here an output variable is chosen and its derivative is obtained ... |
Therefore, the sequence for reverse accumulation is such that the function value is first evaluated and stored during a forward sweep. Then, this is followed by accumulation of the function derivative with respect to each of the intermediate and input variables during the reverse sweep. Hence, at the end of a reverse s... |
The accumulation process is achieved through an associated scalar variable [MATH] (also called adjoint variable ) attached to every vertex. The adjoint [MATH] for the output vertex [MATH] is initially set to [MATH] , while that of each of the remaining [MATH] vertices is set to zero. These initial adjoint values for th... |
[EQUATION] Technically, the reverse mode resembles symbolic differentiation in the sense that one starts with the final result in the form of a formula for the function and then successively applies chain rule down the graph until the independent variables are reached. Recall from the listing for the function values in... |
[EQUATION] In the same way, the derivative of the input vertex [MATH] is: [EQUATION] Because all the vertices with the exception of [MATH] and [MATH] are independent of [MATH] (see the listings in table 4.2 ), their derivative with respect to [MATH] is zero. Hence, the derivative [MATH] reduces to: |
[EQUATION] [EQUATION] The reverse derivatives of all the remaining vertices are accumulated in the above manner and the complete listing is shown in table 4.4 . Notice how the sequence is reversed, i.e., while the values were initially obtained in a forward sweep, the derivatives here are evaluated beginning at the out... |
Notice that seeding the adjoint of the output vertex to [MATH] , (i.e. setting [MATH] ) and evaluating downward sets the adjoints of the input variables [MATH] and [MATH] to be the derivatives of the function with respect to [MATH] and [MATH] respectively. This is where reverse accumulation method tends to supersede th... |
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