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A Variant Constrained Genetic Algorithm for Solving Conditional Nonlinear Optimal Perturbations


doi: 10.1007/s00376-013-2253-6

  • A variant constrained genetic algorithm (VCGA) for effective tracking of conditional nonlinear optimal perturbations (CNOPs) is presented. Compared with traditional constraint handling methods, the treatment of the constraint condition in VCGA is relatively easy to implement. Moreover, it does not require adjustments to indefinite parameters. Using a hybrid crossover operator and the newly developed multi-ply mutation operator, VCGA improves the performance of GAs. To demonstrate the capability of VCGA to catch CNOPS in non-smooth cases, a partial differential equation, which has onoff switches in its forcing term, is employed as the nonlinear model. To search global CNOPs of the nonlinear model, numerical experiments using VCGA, the traditional gradient descent algorithm based on the adjoint method (ADJ), and a GA using tournament selection operation and the niching technique (GA-DEB) were performed. The results with various initial reference states showed that, in smooth cases, all three optimization methods are able to catch global CNOPs. Nevertheless, in non-smooth situations, a large proportion of CNOPs captured by the ADJ are local. Compared with ADJ, the performance of GA-DEB shows considerable improvement, but it is far below VCGA. Further, the impacts of population sizes on both VCGA and GA-DEB were investigated. The results were used to estimate the computation time of VCGA and GA-DEB in obtaining CNOPs. The computational costs for VCGA, GA-DEB and ADJ to catch CNOPs of the nonlinear model are also compared.
    摘要: A variant constrained genetic algorithm (VCGA) for effective tracking of conditional nonlinear optimal perturbations (CNOPs) is presented. Compared with traditional constraint handling methods, the treatment of the constraint condition in VCGA is relatively easy to implement. Moreover, it does not require adjustments to indefinite parameters. Using a hybrid crossover operator and the newly developed multi-ply mutation operator, VCGA improves the performance of GAs. To demonstrate the capability of VCGA to catch CNOPS in non-smooth cases, a partial differential equation, which has onoff switches in its forcing term, is employed as the nonlinear model. To search global CNOPs of the nonlinear model, numerical experiments using VCGA, the traditional gradient descent algorithm based on the adjoint method (ADJ), and a GA using tournament selection operation and the niching technique (GA-DEB) were performed. The results with various initial reference states showed that, in smooth cases, all three optimization methods are able to catch global CNOPs. Nevertheless, in non-smooth situations, a large proportion of CNOPs captured by the ADJ are local. Compared with ADJ, the performance of GA-DEB shows considerable improvement, but it is far below VCGA. Further, the impacts of population sizes on both VCGA and GA-DEB were investigated. The results were used to estimate the computation time of VCGA and GA-DEB in obtaining CNOPs. The computational costs for VCGA, GA-DEB and ADJ to catch CNOPs of the nonlinear model are also compared.
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  • [1] SUN Guodong, MU Mu, ZHANG Yale, 2010: Algorithm Studies on How to Obtain a Conditional Nonlinear Optimal Perturbation (CNOP), ADVANCES IN ATMOSPHERIC SCIENCES, 27, 1311-1321.  doi: 10.1007/s00376-010-9088-1
    [2] WANG Qiang, MU Mu, Henk A. DIJKSTRA, 2012: Application of the Conditional Nonlinear Optimal Perturbation Method to the Predictability Study of the Kuroshio Large Meander, ADVANCES IN ATMOSPHERIC SCIENCES, 29, 118-134.  doi: 10.1007/s00376-011-0199-0
    [3] Bin MU, Juhui REN, Shijin YUAN, Rong-Hua ZHANG, Lei CHEN, Chuan GAO, 2019: The Optimal Precursors for ENSO Events Depicted Using the Gradient-definition-based Method in an Intermediate Coupled Model, ADVANCES IN ATMOSPHERIC SCIENCES, 36, 1381-1392.  doi: 10.1007/s00376-019-9040-y
    [4] JIANG Zhina, 2006: Applications of Conditional Nonlinear Optimal Perturbation to the Study of the Stability and Sensitivity of the Jovian Atmosphere, ADVANCES IN ATMOSPHERIC SCIENCES, 23, 775-783.  doi: 10.1007/s00376-006-0775-x
    [5] Sung Hyup YOU, Yong Hee LEE, Woo Jeong LEE, 2011: Parameterization and Application of Storm Surge/Tide Modeling Using a Genetic Algorithm for Typhoon Periods, ADVANCES IN ATMOSPHERIC SCIENCES, 28, 1067-1076.  doi: 10.1007/s00376-011-0113-9
    [6] FANG Changluan, ZHENG Qin, WU Wenhua, DAI Yi, 2009: Intelligent Optimization Algorithms to VDA of Models with on/off Parameterizations, ADVANCES IN ATMOSPHERIC SCIENCES, 26, 1181-1197.  doi: 10.1007/s00376-009-8084-9
    [7] QIN Xiaohao, MU Mu, 2014: Can Adaptive Observations Improve Tropical Cyclone Intensity Forecasts?, ADVANCES IN ATMOSPHERIC SCIENCES, 31, 252-262.  doi: 10.1007/s00376-013-3008-0
    [8] XIANG Jie, LIAO Qianfeng, HUANG Sixun, LAN Weiren, FENG Qiang, ZHOU Fengcai, 2006: An Application of the Adjoint Method to a Statistical-Dynamical Tropical-Cyclone Prediction Model (SD–90) II: Real Tropical Cyclone Cases, ADVANCES IN ATMOSPHERIC SCIENCES, 23, 118-126.  doi: 10.1007/s00376-006-0012-7
    [9] Zhenhua HUO, Wansuo DUAN, Feifan ZHOU, 2019: Ensemble Forecasts of Tropical Cyclone Track with Orthogonal Conditional Nonlinear Optimal Perturbations, ADVANCES IN ATMOSPHERIC SCIENCES, 36, 231-247.  doi: 10.1007/s00376-018-8001-1
    [10] WANG Zhi, GAO Kun, 2006: Adjoint Sensitivity Experiments of a Meso- -scale Vortex in the Middle Reaches of the Yangtze River, ADVANCES IN ATMOSPHERIC SCIENCES, 23, 267-281.  doi: 10.1007/s00376-006-0267-z
    [11] SUN Guodong, MU Mu, 2012: Inducing Unstable Grassland Equilibrium States Due to Nonlinear Optimal Patterns of Initial and Parameter Perturbations: Theoretical Models, ADVANCES IN ATMOSPHERIC SCIENCES, 29, 79-90.  doi: 10.1007/s00376-011-0226-1
    [12] H. Kurtulus OZCAN, Erdem BILGILI, Ulku SAHIN, O. Nuri UCAN, Cuma BAYAT, 2007: Modeling of Trophospheric Ozone Concentrations Using Genetically Trained Multi-Level Cellular Neural Networks, ADVANCES IN ATMOSPHERIC SCIENCES, 24, 907-914.  doi: 10.1007/s00376-007-0907-y
    [13] SUN Guodong, MU Mu, 2013: Using the Lund-Potsdam-Jena Model to Understand the Different Responses of Three Woody Plants to Land Use in China, ADVANCES IN ATMOSPHERIC SCIENCES, 30, 515-524.  doi: 10.1007/s00376-012-2011-1
    [14] CHEN Boyu, MU Mu, 2012: The Roles of Spatial Locations and Patterns of Initial Errors in the Uncertainties of Tropical Cyclone Forecasts, ADVANCES IN ATMOSPHERIC SCIENCES, 29, 63-78.  doi: 10.1007/s00376-011-0201-x
    [15] SUN Guodong, MU Mu, 2011: Response of a Grassland Ecosystem to Climate Change in a Theoretical Model, ADVANCES IN ATMOSPHERIC SCIENCES, 28, 1266-1278.  doi: 10.1007/s00376-011-0169-6
    [16] WANG Bo, and HUO Zhenhua, 2013: Extended application of the conditional nonlinear optimal parameter perturbation method in the Common Land Model, ADVANCES IN ATMOSPHERIC SCIENCES, 30, 1213-1223.  doi: 10.1007/s00376-012-2025-8
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Manuscript received: 12 October 2012
Manuscript revised: 13 January 2013
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A Variant Constrained Genetic Algorithm for Solving Conditional Nonlinear Optimal Perturbations

  • 1. Institute of Science, PLA University of Science and Technology, Nanjing 211101
  • 2. State Key Laboratory of Numerical Modeling for Atmospheric Sciences and Geophysical Fluid Dynamics, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing 100029
  • 3. Troop 94906, People’s Liberation Army, Suzhou 215157

Abstract: A variant constrained genetic algorithm (VCGA) for effective tracking of conditional nonlinear optimal perturbations (CNOPs) is presented. Compared with traditional constraint handling methods, the treatment of the constraint condition in VCGA is relatively easy to implement. Moreover, it does not require adjustments to indefinite parameters. Using a hybrid crossover operator and the newly developed multi-ply mutation operator, VCGA improves the performance of GAs. To demonstrate the capability of VCGA to catch CNOPS in non-smooth cases, a partial differential equation, which has onoff switches in its forcing term, is employed as the nonlinear model. To search global CNOPs of the nonlinear model, numerical experiments using VCGA, the traditional gradient descent algorithm based on the adjoint method (ADJ), and a GA using tournament selection operation and the niching technique (GA-DEB) were performed. The results with various initial reference states showed that, in smooth cases, all three optimization methods are able to catch global CNOPs. Nevertheless, in non-smooth situations, a large proportion of CNOPs captured by the ADJ are local. Compared with ADJ, the performance of GA-DEB shows considerable improvement, but it is far below VCGA. Further, the impacts of population sizes on both VCGA and GA-DEB were investigated. The results were used to estimate the computation time of VCGA and GA-DEB in obtaining CNOPs. The computational costs for VCGA, GA-DEB and ADJ to catch CNOPs of the nonlinear model are also compared.

摘要: A variant constrained genetic algorithm (VCGA) for effective tracking of conditional nonlinear optimal perturbations (CNOPs) is presented. Compared with traditional constraint handling methods, the treatment of the constraint condition in VCGA is relatively easy to implement. Moreover, it does not require adjustments to indefinite parameters. Using a hybrid crossover operator and the newly developed multi-ply mutation operator, VCGA improves the performance of GAs. To demonstrate the capability of VCGA to catch CNOPS in non-smooth cases, a partial differential equation, which has onoff switches in its forcing term, is employed as the nonlinear model. To search global CNOPs of the nonlinear model, numerical experiments using VCGA, the traditional gradient descent algorithm based on the adjoint method (ADJ), and a GA using tournament selection operation and the niching technique (GA-DEB) were performed. The results with various initial reference states showed that, in smooth cases, all three optimization methods are able to catch global CNOPs. Nevertheless, in non-smooth situations, a large proportion of CNOPs captured by the ADJ are local. Compared with ADJ, the performance of GA-DEB shows considerable improvement, but it is far below VCGA. Further, the impacts of population sizes on both VCGA and GA-DEB were investigated. The results were used to estimate the computation time of VCGA and GA-DEB in obtaining CNOPs. The computational costs for VCGA, GA-DEB and ADJ to catch CNOPs of the nonlinear model are also compared.

1 Introduction
  • The nonlinear influence should not be ignored when studying the predictability and stability of atmospheric and oceanic movements (Mu2000; Ding2007; Ding2010; Li2011, 2012). Instead of using a linear singular vector, Mu et al. (2003) proposed the notion and theory of using a conditional nonlinear optimal perturbation (CNOP), which represents a kind of initial perturbation that has the largest nonlinear evolution at the end of the concerned time window. Physically, a CNOP describes the initial error that satisfies a certain constraint and yields the largest prediction error at the prediction time. If this initial error exists in practical predictions, it is possible to improve the forecast by filtering out the CNOP errors with a data assimilation method or target observation method. In fact, it has been verified that CNOP errors occur in real predictions of ENSO, and thus ENSO forecast skill could be greatly improved by filtering out these errors [Duan and Wei, 2012]. The CNOP method has attracted increasing attention and has been applied successfully to ENSO predictability (Duan et al., 2004;Duan and Mu, 2006; Duan et al., 2008; Duan et al., 2009), the sensitivity and stability of oceanic thermocline circulation (Mu et al., 2004), the nonlinear behaviors of oceanic double-gyre circulation and baroclinic unstable flow [Sun et al., 2005], and the generation of the onset of perturbation triggering blocking (MuJ et al., 2009; Jiang et al., 2010). Its effectiveness has also been tested in studies of ensemble prediction and targeted observations (Mu et al., 2009; Jiang et al., 2009). Recently, [Duan, et. al.] investigated the initial anomalies of CNOPs and linear singular vectors (LSVs) in the Zebiak-Cane model [Zebiak and Cane, 1987], concluding that CNOPs can act as the optimal precursor for EI Niño events.

    Calculating CNOPs requires a constrained nonlinear optimization algorithm. As we know, the main optimization algorithms used to search CNOPs, such as the spectral projected gradient method (SPG) and sequential quadratic programming (SQP), are based on the gradient descent method, and the gradient is often provided by the associated adjoint (ADJ) model (Duan2004; Duan2008; MuJ2009; Mu2009; Jiang2010). However, adjoint technology needs a highly-qualified environment (i.e., the nonlinearity of the governing equation is not too strong, and the objective function is differentiable with respect to the optimization variables) to be carried out. Moreover, in fact, not all models have adjoint models, while it is quite tedious and time-consuming to write an adjoint model of a complex model. With the gradual improvement of atmosphere and ocean models, an increasing amount of parameterized physical processes are being added into models on an ongoing basis. This, in one sense, helps to more precisely describe the movements of the atmosphere or oceans; while, from a different perspective, it also brings non-smoothness in the form of “on-off” switches into the models. Such “on-off ” switches aggravate the nonlinearity of models, consequently weakening the effectiveness of algorithms based on the adjoint technique (Xu1996, 1997; Xu1998; Mu2005; Zheng2006). In order to avoid this problem, only the dry energy norm is used to identify the sensitive areas in targeted observations for tropical cyclone prediction, without any moist physical process in the adjoint system [Mu et al., 2009]. The aim of the present study is to explore an optimization algorithm that has the ability to capture global CNOPs effectively in non-smooth and strongly nonlinear situations.

    Genetic algorithms (GAs) were introduced by John Holland in 1975 [Holland, 1975]. Unlike traditional optimization methods based on gradient descent, GAs are independent of the actual form of the model. Moreover, the roughness of the objective function has almost no impact on the performance of GAs, and thus they are more powerful than traditional gradient descent optimization methods when the objective function is non-differentiable or even discontinuous [Zheng et al., 2012].

    In a GA, there are three basic genetic operators, i.e., selection, crossover and mutation operators. Proper configuration of these operators can improve the performance of the GA searching the optimum. However, for different types of optimization problems, genetic operators with specific configuration can impact upon the effectiveness of the GA in different ways. To solve complex optimization problems quickly and accurately using this approach, it is necessary to develop a competent GA.

    When GAs are used to solve constrained optimization problems, how to deal with constraint conditions properly is very important. Currently, the most widely used method is the penalty function approach (Homaifar1994; Joines1994; Michalewicz1995; MichalewiczS1996). Fang and Zheng (2009) applied the penalty function method to an ideal “on-off” simplified model to investigate the effectiveness of GAs in capturing CNOPs. Although the penalty function method is simple and easily implemented, the optimal solution of the new fitness function with an additional penalty term depends on the choice of penalty parameters. To find out what penalty parameter will drive the search towards the feasible region, one usually has to conduct many numerical experiments with different parameter values. Additionally, the inclusion of the penalty term distorts the objective function: if the penalty parameters are too small, the optimum of the objective function with the additional penalty term may not be close to the true constrained optimum; but if the penalty parameters are too large, the objective function may have artificially local optimal solutions. [Deb, 2000] developed a constraint handling method based on the tournament selection mechanism and niching strategy for GAs (known as the GA-DEB). Its advantage is that it does not require any penalty parameters.

    The present study focuses on an optimization algorithm that has the ability to capture global CNOPs effectively in non-smooth and strongly nonlinear situations, and can also be easily implemented. Benefiting from Deb’s (2000) constraint handling method, and in view of the characteristics of the constraint conditions of a CNOP (i.e., the geometric characteristics of a ball), we propose a new constrained optimization algorithm —a variant constrained genetic algorithm (VCGA) — to solve CNOPs. The VCGA not only overcomes the uncertainty of adjusting penalty parameters in the penalty function method, but also shows better performance than either ADJ or GA-DEB in searching global CNOPs in non-smooth and strong nonlinear problems.

    The remainder of the paper is organized as follows. The nonlinear model adopted as the governing equation is described in the following section (section 2), together with the basic concept of a CNOP. In section 3, after a summary account of the common constraint handling methods in GAs, the genetic strategies and constraint handling methods used in the VCGA are introduced in detail. Section 4 is devoted to comparative numerical experiments of the three optimization schemes, i.e., ADJ, GA-DEB and VCGA, and a statistical analysis of experimental results. Finally, a discussion and conclusions are presented in section 5.

2 Nonlinear models and CNOPs
  • The following idealized nonlinear model used in [Zheng et al., 2012] is adopted as the governing equation:

    where q(t,l) ≥0 denotes the specific humidity; qs is the saturation specific humidity, called “ threshold ”; t is the time variable; l stands for either horizontal variable, x or y, or the vertical variable z; ζ=ζ(t,l), the velocity in the l direction, is a given continuous function with all first order partial derivatives being continuous, and ζ(t,l)>0,l≠L, and ζ(t,L)=0; the initial specific humidity q0(l) is continuously differentiable in [0,L] satisfying dq0/dl<0; the constant G denotes the source term due to the parameterization process and the function A(t) of time t stands for the source term due to other physical processes. In order to ensure the “ on-off ” switches are triggered repeatedly at some space grid points during the period of time concerned, it is assumed that q0(l)<qs and A(0)-G>0.

    By using the windward format and the conventional numerical discretization of “ on-off ” processes, Eq. (1) can be discretized as follows:

    q0,i=q0(li),i=0,1, ···,l

    qk,0=qk-1,0+[A(tk-1)-GH(qk-1,0-qs)] Δt,1 k K

    +[A(tk-1) -GH(qk-1,i -qs)] Δt,1kK;liL,

    where Δt denotes the time step; tk=kΔt; Δl is the space step; li=iΔl; k is the time level; i is the space grid point; L+1=(L/Δl)+q is the number of space grid points; and K=T/Δt is the number of time levels in integration. The following Courant-Friedrichs-Levy condition

    is always satisfied in the numerical experiments conducted in this study.

  • Assuming the model that simulates the motions of the atmosphere or ocean is as follows:

    where w(x,t)=(w1(x,t),w2(x,t), ···,wm(x,t))T, B is the nonlinear operator; w0(x) is the initial state; Ω is a domain of Rn;(x,t)∈Ω×[0,T]. x=(x1,x2, ···,xn) and t are the spatial and temporal variables, respectively; t=0 is the initial time; and t=T with T<+∞ is a future time. If Mt denotes the propagator of Eq. (2) from 0 to t(0 t T), then the solution of Eq. (2) at time t, which is referred to as the reference state hereafter, just as in [Mu et al., 2003], can be given by

    w(x,t)=Mt(w0), (4)

    Superposing an initial perturbation upon w0(x) and denoting as the nonlinear evolution of the initial perturbation with time t, we have:

    With a chosen norm PP measuring perturbation , the objective function is defined as follows:

    The initial perturbation satisfying

    is called a CNOP at the future time T, where δ>0 is the radius of the ball constraining the initial perturbations and is specified according to the requirements of real problems.

3 Applying GAs to capture CNOPs
  • GAs are global optimization search algorithms. Because of their group search strategy and computational method being independent of gradient information, GAs are more popular than traditional deterministic search methods in dealing with constrained optimization problems. In particular, when the nonlinear propagator Mt(w0) is non-differentiable or discontinuous with respect to the initial reference state w0, there are three advantages to applying GAs to capture CNOPs: (1) the probability of GAs to capture global CNOPs is higher than other deterministic optimization methods; (2) the constraint conditions of CNOPs are easy to implement due to the flexible constraint handling method for GAs; and (2) the computation efficiency of GAs can be enhanced by designing proper parallel schemes for high-dimension CNOPs.

    When applying a GA to solve an optimization problem, it is required to operate encoding, population initialization, population evaluation and configuration of genetic operators. Fig. 1 illustrates the basic steps included in a GA.

  • A constrained optimization problem is usually written in the following form:

    minf(x), x=(x1,x2, ···,xn) ∈Rn (8)

    gj(x) ≥0, j=1,2,3, ···,J (9)

    hj(x) = 0 , j=J=1, ···,P, (10)

    xi,lo ≤ xi ≤ xi,up, 1≤i≤n (11)

    Figure 1.  Flow chart showing the basic steps included in a GA.

    Where Ω= {x = (x1,x2, ,···,xn) ∈ Rn | gj(x) > 0, 1≤j≤J; hj(x) = 0, J+1≤j≤p; xi,lo≤xi≤xi,up, 1≤i≤n} is the feasible region; x∈Ωis the decision vector; J is the number greater-than-or-equal-to type inequality constraints; (p-J) is the number of equality constraints; f(x) is the objective function; gj(x) is the jth inequality constraint; hj(x) is the jth equality constraint; xi,lo,xi,up are the lower and upper bounds, respectively, of the ith variable xi.

    In general, equality constraints in Eq. (9) are handled by converting them into the following inequality constraints:

    i-|h(x)| > 0, (12)

    where t≥0 is the tolerance value of the equality constraints—usually a small positive number. After converting the equality constraints into inequality constraints, there will be p inequality constraints in the constrained optimization problem, Eqs. (7)-(10).

    There are many constraint handling methods used in GAs, compared with other methods, but the penalty function method is the most widely adopted. Its main idea is to penalize infeasible solutions, i.e., to solve an unconstrained optimization problem using the following modified objective function, called the penalty fitness function:

    F(x) = f (x)+penalty(x),

    where penalty(x) is zero when x∈Ω; otherwise, it is positive. Since the degree to which the individual x violates the jth constraint can be measured by

    [Homaifar et al., 1994] devised the following penalty fitness function:

    where is the penalty term and Rj(j=1,2,···,p) is the penalty coefficient. Since none of Rj changes with the iteration number (i.e., the generation number), this method is called the static penalty function method. [Joines and Houck, 1994] proposed the following dynamic penalty method:

    where v is the iteration number; and C, α, βare constants requiring adjustment. More detail about penalty functions can be found in [Michalewicz and Schoenauer, 1996], and references therein. Although penalty function methods have received wide attention because of their simplicity and ease of implementation, they still have some disadvantages. The biggest drawback is that extensive experimentation is required for setting up appropriate parameters needed to define the penalty function. To deal with this issue, [Deb, 2000] developed a constraint handling method. This method uses the tournament selection mechanism and niching strategy in the selection operation of the GA. The comparison criteria of the tournament selection are: (1) feasible solutions are always superior to infeasible solutions; (2) when two feasible solutions are compared, the one having the better objective function will be chosen; and (2) when two infeasible solutions are compared, the one having the smaller constraint violation will be selected. Maintaining diversity among feasible solutions is an important task for GAs, which was achieved by [Deb, 2000] through use of the niching strategy in the tournament selection operator.

    However, each method has its own applicability and there is still no general method that can deal with all kinds of constraints, as stated by Gregory (1995):

    “It is unrealistic to expect to find one general nonlinear programming code that is going to work for every kind of nonlinear model. Instead, you should try to select a code that fits the problem you are solving.”

  • When carrying out the genetic operation on parents to generate offspring, we need to implement selection, crossover, mutation operators and/or the elitist reservation strategy. Proper genetic operators will make GAs more effective. In trying to search the global CNOP in a non-smooth and strongly nonlinear case, an effective constrained optimization algorithm, the VCGA, was developed in the present reported work and its genetic operations are as follows.

    3.2.1 Selection operation

    The selection operation in the VCGA uses a tournament selection operator. Specifically, two individuals are picked at random from the current population and are compared based on the following comparison criteria:

    (1) When both comparative individuals are feasible solutions, the one with the better objective function is preferred.

    (2) When there is any infeasible solution among two comparative individuals, firstly pull the infeasible solution to the edge of the spherical constraints — that is, substitute individual x with y=δx/PxP if it is an infeasible solution, where δ>0 is the radius of the ball constraining the initial perturbations—and then perform comparison criterion (1).

    3.2.2 Crossover operation

    [Herrera et al., 2005] analyzed the performance of BLX-αand indicated that, since it randomly generates offspring genes in the neighborhood of parent genes, the operator has a good global searching ability. Thus, offspring individuals can retain good diversity. [Deb and Agrawal, 1995] discussed the particularities of the simulation binary crossover (SBX) operator and pointed out that, since the range of offspring generated from SBX is in proportion to that of the parents, and the offspring solution near the parent solutions can be generated with higher probability, SBX not only has the ability to be adaptable, but can also attain the optimal solutions of a multimodal objective function. Motivated by these results, we chose BLX-αor SBX randomly to each crossover operation in VCGA.

    3.2.3 Mutation operation

    A GA is an algorithm that simulates biological evolution. During evolution, the genes of an organism may be subject to several different kinds of mutation leading up to the final state. This inspired us to devise a new mutation operator, which we named — and is referred to as hereafter — the multi-ply mutation operator. The way that the multiply mutation operator performs is as follows.

    For each gene xi(1≤I≤n) of an individual x=(x1,x2,···,xn), the multi-ply mutation operator carries out the non-uniform mutation, parameter-based mutation, boundary mutation and uniform mutation in turn according to mutation probabilities Pm,1,Pm,2,Pm,3 and Pm,4 respectively. The procedure for operating the multi-ply mutation operator is shown in Fig. 2, in which ri,i=1,2,3,4 are uniformly distributed random numbers in [0,1], and Pm,i = 1,2,3,4 is the mutation probability.

    3.2.3.1 The non-uniform mutation

    The non-uniform mutation operation used in our multi-ply mutation operator was proposed by [Michalewicz, 1996]. Its execution procedure is as follows.

    For each parent solution x=(x1,x2,···,xn) in a population of the Xth generation, create a new individual y=(y1,y2,···,yn) in the following way:

    yi=xi + Δ(X,xi,up - xi)or yi = xi - Δ(X,xi - xi,lo), 1≤i≤n (16)

    where Δ(X,y)=yr(1 - X/Tmax) b; r is a uniform random number from [0,1]; Tmax is the maximal evolution generation number; b is a parameter used to determine the degree of non-uniformity; and b=2 (in this study).

    The non-uniform mutation operator, which carries out a uniform search in the early part of the evolutionary process and a local search in the later part, has slight adjustment ability. It is one of the most effective mutation operators [Herrera et al., 1998].

    Figure 2.  Flow chart showing multi-ply mutation operation in VCGA.

    3.2.3.2 Parameter-based mutation

    The following parameter-based mutation operation was presented by [Deb, 2000] and is used in the multi-ply mutation operator.

    Let x=(x1,x2,···,xn) be an individual, and for each gene xi(i=1,2,···,n), both lower and upper boundaries xi,lo and xi,up are specified.

    Step (1): Generate a uniformly distributed random number μ in [0,1];

    Step (2): Calculate the parameter as follows:

    where ηm is the distribution index for mutation with an arbitrary non-negative value, and δ=min((xi-xi,lo),(xi,up-xi)/(xi,up-xi,lo));

    Step (2): Calculate the mutated individual as follows:

    where Δmax=xi,up-xi,lo is the maximum perturbation allowed in x=(x1,x2,···,xn).

    3.2.3.3 Boundary mutation

    The boundary mutation operation is as follows:

    Step (1): Obtain an integer k from the integer set {1,2,···n} with the same probability;

    Step (2): Generate a uniformly distributed random number u in [0,1];

    Step (2): Generate a new individual y=(y1,y2,···yn) from the individual x=(x1,x2,···xn):

    3.2.3.4 Uniform mutation

    The uniform mutation operation is as follows.

    Step (1): Obtain an integer k from the integer set {1,2,···n} with the same probability.

    Step (2): Generate a new individual y=(y1,y2,···yn) from the individual x=(x1,x2,···,xn)

    where r is a uniformly distributed random number in [xi,lo,xi,up].

    3.2.3.4 Elitist reservation strategy

    Figure 3.  Flow chart showing elitist reservation strategy operation in VCGA.

    Since the fittest may be lost in crossover and mutation operations, the elitist reservation strategy is employed in the VCGA to avoid this possibility. The detailed steps involved in the operation of the elitist reservation strategy are shown in Fig. 3, in which M is the population size; xi(1),xi(2),···xi(M) is the parent generation; xx+1(1),xx+1(2),···xx+1(M) is the current generation; Ji(m) is the objective function value of the individual xi(m)(m=1,2,··M·); and J(x+1),avg is the average of individual objective function values of the current generation. To operate the elitist reservation strategy shown in Fig. 3 for a generation, its individuals must first be arrayed in their decreasing objective function values.

4 Numerical experiments for solving CNOPs and their results
  • In the numerical experiments for solving CNOPs, the discrete objective function in Eq. (6) is

    where is an initial perturbation vector; is the nonlinear evolution of at time T=KΔt;MT is the nonlinear propagator of Eq. (2); and q0=(q0,0,q0,1, ···,q0,I) is a given initial reference state.

    In order to verify the effectiveness of the VCGA for solving CNOPs in non-smooth and strongly nonlinear situations, numerical experiments for solving CNOPs were conducted respectively with three optimization methods, i.e., ADJ, GA-DEB and VCGA. In ADJ, the constrained optimization algorithm was the spectral projected gradient method, version 2 (SPG2) [Birgin et al., 2000] and the required gradient information was provided by integrating backward the associated adjoint model of Eq. (2). In GA-DEB, the optimization algorithm used was developed by [Deb, 2000], which is a GA based on the tournament selection mechanism and niching strategy.

    Since the initial specific humidity q0(l)>0 is required to be continuously differentiable and satisfy q0(l)<qs,dq0/dl<0 in [0,L], we therefore set the range of initial perturbation as

    When CNOPs are used to solve practical problems, we need to select a suitable constraint radius δ. Usually, δ is an estimation of the analytical field error variance [Mu et al., 2009]. In [Mu and Zhang, 2006], the constraint radiusδranges from 1.8%-2% of the L2 norm of the initial reference state when an ideal test with respect to a 2D quasi-geostrophic model is carried out. Accordingly, besides Eqs. (20)-(22), the initial perturbation in our numerical experiments satisfied

  • In order to intuitively display the performance of the three optimization algorithms in solving CNOPs, similar numerical experiments as those in [Fang and Zheng, 2009] were first conducted, i.e., the initial perturbation had the form , where i1,i2 were selected randomly from the set {0,1,···I}. The parameters of Eq. (2) adopted in the numerical experiments were: A(t)=8-11t;qs=0.58; ζ(t,l)=(1+t)(1-l);I=20;K=100; Δt=0.01; andΔl=0.05. The related parameters in the GAs were: population size M=40; crossover probability was 0.85; and mutation probabilities were Pm,1=0.01,Pm,2=Pm,3=Pm,4=0.001. Wherever the niching strategy was used, we had the critical individual numberθc=15, and the critical distance . The distribution index for SBX wasηc=2, for the parameter-based mutation it was ηm=100, and the maximum evolutionary generation was 100. The initial perturbations for ADJ or the initial populations for both GA-DEB and VCGA were randomly generated in the assigned range—that is, they needed to satisfy Eqs. (20)-(22). The initial reference state was q0(l)=0.28-0.15sin(π/2).

    When physical parameterization processes were removed, i.e., g=0 in Eq. (2), the numerical experiment results for the two grid points i1=5,i2=8 selected to superpose perturbations are shown in Fig. 4. In Fig. 4a, the two green points pointed to by arrows are the locations of CNOPs. The optimal solutions attained by ADJ, GA-DEB and VCGA are shown in Fig. 4b, which demonstrates that when there is no influence of “on-off ” switches, all three optimization algorithms can obtain the global CNOP accurately.

    Figure 4.  In smooth cases (g=0), the logarithm of the objective function value with respect to the 6th and 9th grid point perturbations, the CNOP (a) and the optimal solutions captured by ADJ, GA-DEB and VCGA (b). For the sake of clarity, both the x-axis and y-axis are amplified by 103.

    Figure 5.  In a non-smooth case (g=7), the logarithm of the objective function value with respect to the 5th and 7th grid point perturbations: (a) shows the optimal solution tracked by ADJ; (b) and (c) show those by GA-DEB and VCGA, respectively. For the sake of clarity, both the x-axis and y-axis are amplified by 103.

    When physical parameterization processes were included, the correlative parameters in the numerical experiments were taken as g=7,i1=4, and i2=6. Since the “ on-off ” switches in Eq. (2) were triggered repeatedly, and although only two components of the initial reference state were perturbed, the behavior of was very poor (Figs. 5a-c). The CNOPs and the optimal solutions obtained by ADJ, GA-DEB and VCGA are shown in Fig. 5, in which the green lines with arrows illustrate the optimization trace of the three optimization methods. A comparison among the optimization results of the three optimization methods demonstrates that both VCGA and GA-Deb can still effectively capture CNOPs in non-smooth cases, while ADJ fails to do well. In addition, although GA-DEB can capture the CNOP, it requires a lot of iteration before it reaches the optimal solution (Fig. 5b). In contrast, VCGA can attain the CNOP with less iteration (Fig. 5c).

  • We saw in the above section that, for 2D CNOPs, VCGA has a strong ability to catch them, even in non-smooth cases. However, we do not know whether or not the performance of VCGA will deteriorate with an increase in the dimensions of CNOP, which is important to investigate because the dimensions of real problems will be very high. In this section, we performed numerical experiments for the three optimization schemes (ADJ, GA-DEB and VCGA) with general initial perturbations . All parameters used in the experiments were the same as those reported in section 4.1, except that the population size was 60 and the maximum iteration was 180.

    For each optimization scheme, we performed 200 runs from different initial perturbations (ADJ) or initial populations (GA-DEB and VCGA). In smooth cases (g=0), all three optimization schemes captured CNOPs effectively. However, in non-smooth situations, where “on-off” switches are triggered repeatedly (g=7), the numerical experiment results showed that 200 CNOPs captured by VCGA were all global, while only 84% of 200 results generated by GA-DEB were global (Fig. 6). Compared with GA-DEB, VCGA displayed a stronger ability for catching CNOPs. Since the associated adjoint model could not provide correct gradient information, ADJ often fell into local optimal solutions in the 200 numerical experiments. Its ability to capture global CNOPs was the poorest among the three optimization algorithms.

    Figure 6.  The logarithm of the objective function value 1gJ corresponding to the optimal results of the 200 experiments by the three algorithms in non-smooth cases.

    In real applications, not only the accuracy of the optimization solution must be considered, but also the computational cost. For a GA, the population size and evolutionary generation are two main factors that influence its computational cost. To evaluate the computational costs of GA-DEB and VCGA in obtaining CNOPs fairly and objectively, their performances in solving the CNOPs were tested for different population sizes. The convergence criterion was no longer determined by a pre-specified maximum evolutionary generation; instead, it was determined by a specified threshold value that indicated how close the best individual of the current generation was to the best ones of the previous several generations (taken as 15 generations in the numerical experiments). When the differences among the best individuals of adjacent generations were smaller than the specified threshold value, the evolving process was terminated. Population sizes of 4, 12, 22, 42, 64, 84 and 96, which were approximately 20%, 50%, 100%, 200%, 300%, 400% and 450% of the dimensions of the optimization variables, were adopted. The threshold value used to terminate the evolving process was set as 10-4 in the test numerical experiments. For both VCGA and GA-DEB, and each given population size, a total of 200 numerical experiments were performed in smooth and non-smooth cases. The results were statistically analyzed, as presented in Tables 1 and 2.

    From Tables 1 and 2, it can clearly be seen that the optimal population size of VCGA was about 12 in both smooth and non-smooth cases. However, for GA-DEB, its optimal population size was larger than 96. Furthermore, the computation times consumed respectively by VCGA with a population size of 12, and GA-DEB with a population size of 96, were compared with that of ADJ. Table 3 illustrates the average computation time consumed by the three methods in their respective 200 numerical experiments. We can see that, in smooth cases, ADJ consumed less computation time than the other two schemes—about 1/23 of that used by VCGA, and less than 1/51 of that used by GA-DEB. However, in non-smooth cases, VCGA consumed the least computation time. Moreover, the difference between the computation time of VCGA in smooth and non-smooth cases was very small, while it was very large for ADJ. Therefore, roughness has almost no impact on the performance of VCGA, but has a very large influence on ADJ.

  • The following five initial reference states

    were used to further test the performance of the three optimization schemes for solving CNOPs in non-smooth cases. For each, VCGA, GA-DEB and ADJ were respectively used to catch its CNOP, and the population sizes used in VCGA and GA-DEB were 22 and 96, respectively. Each numerical experiment was performed 200 times from different initial perturbations (ADJ) or initial populations (GA-DEB and VCGA). Table 4 demonstrates the average of the logarithm of the objective function values of the CNOPs generated by each optimization scheme for each of the initial reference states given above. Table 5 indicates the average computation time corresponding to Table 4. Tables 4 and 5 show that, in the non-smooth case, the accuracy of the CNOP attained by VCGA was the highest. Moreover, the computational cost it consumed was much lower than that of GA-DEB. Table 6 presents the results of statistical analyses of standard variances of the objective function values for the 200 CNOPs attained in each numerical experiment. It can be seen that the robustness of VCGA was much stronger than both GA-DEB and ADJ, demonstrating the greater effectiveness of both the configuration of genetic operators and the constraint handling method adopted in VCGA compared to Deb’s (2000) method.

5 Summary and discussion
  • During the last decade, CNOPs have been applied in more and more fields. So, it is of vital importance to develop optimization algorithms to obtain CNOPs effectively in the context of roughness and strong nonlinearity. The optimization algorithm based on the gradient descent method requires gradient information on the objective function, which has been successfully solved by ADJ in smooth cases. However, as atmospheric and oceanic models become closer to reality, many complex physical processes begin to merge into models in the form of parameterization. This makes models possess strong nonlinearity, and discontinuous “on-off” processes, whose influence on the traditional ADJ method has been discussed in many studies (Zou et al., 1993; Xu, 1996; Zou, 1997; Mu and Wang, 2003; Fillion and Belair, 2004; Mu and Zheng, 2005).

    For a nonlinear model, capturing its CNOPs requires a constrained nonlinear optimization algorithm. Although GAs have more power than traditional gradient descent approaches when dealing with complex non-smooth and nonlinear optimization problems, there is no systematic way to address the issues of constraints for GAs. The penalty function approach is the most popular method to handle constraint conditions, but it involves a number of penalty parameters and their configuration depends on trial and error. Additionally, the performance of the penalty function approach is dependent on the case. Therefore, it is worth developing a handling constraint method for GAs in order to capture CNOPs accurately.

    In view of the form of the constraint conditions of CNOPs, this paper has introduced a kind of simple tournament selection mechanism to handle constraints, i.e., (1) when both two comparative individuals are feasible solutions, the one with the better objective function is preferred; and (2) when there is any infeasible solution among the two comparative individuals, firstly pull the infeasible solution to the edge of the spherical constraints, and then perform comparison criterion (1). Furthermore, the hybrid crossover operator and the newly developed multi-ply mutation operator were employed in VCGA in order to improve the performance of the GA. The numerical experiments conducted showed that, in non-smooth environments, GAs have a better ability to search global CNOPs than the gradient descent algorithm based on ADJ. In addition, the performance of a GA can be improved greatly provided the genetic operators (selection operator, crossover operator and mutation operator) are properly configured.

    In non-smooth and strongly nonlinear situations, the effectiveness of VCGA in searching global CNOPs has been tested using a simple model. Although time consumption may be a limitation of VCGA, it does not use ADJ (writing an adjoint model for a real complex prediction system is burdensome). Furthermore, it can deal with non-smooth objective functions. However, for actual atmospheric or oceanic models, considering their more complex nonlinear processes and higher dimensions of state variables, further work is still needed test whether or not VCGA would work well. This will be our next research direction.

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