优化函数 （Optimization Function）

以下是一个使用贝叶斯优化的示例函数的Matlab代码： ```matlab function [xopt,fopt]=bayesopt_fun(f,x0,lb,ub,opts) % BAYESOPT_FUN: Bayesian optimization of a function % [XOPT,FOPT]=BAYESOPT_FUN(F,X0,LB,UB,OPTS) finds the minimum of a % function F using Bayesian optimization. X0 is the initial guess, % LB and UB are the lower and upper bounds of the variables, and OPTS % is an options structure created using BAYESOPT_OPTIONS. The function % F should take a vector of variables as input and return a scalar % output. % % Example usage: % f=@(x) sin(3*x) + x.^2 - 0.7*x; % opts=bayesopt_options('AcquisitionFunctionName','expected-improvement-plus'); % [xopt,fopt]=bayesopt_fun(f,0,0,1,opts); % % See also BAYESOPT_OPTIONS. % Check inputs narginchk(4,5); if nargin < 5, opts=bayesopt_options(); end assert(isa(f,'function_handle'),'F must be a function handle'); assert(isvector(x0) && isnumeric(x0),'X0 must be a numeric vector'); assert(isvector(lb) && isnumeric(lb),'LB must be a numeric vector'); assert(isvector(ub) && isnumeric(ub),'UB must be a numeric vector'); assert(all(size(x0)==size(lb)) && all(size(x0)==size(ub)), ... 'X0, LB, and UB must have the same size'); opts=bayesopt_options(opts); % ensure opts has all fields % Initialize X=x0(:); % column vector Y=f(X); n=numel(X); Xbest=X; Ybest=Y; fmin=min(Y); fmax=max(Y); % Loop over iterations for i=1:opts.MaxIterations % Train surrogate model model=fitrgp(X,Y,'Basis','linear','FitMethod','exact', ... 'PredictMethod','exact','Standardize',true, ... 'KernelFunction',opts.KernelFunction,'KernelParameters',opts.KernelParameters); % Find next point to evaluate if strcmp(opts.AcquisitionFunctionName,'expected-improvement-plus') % Use expected improvement with small positive improvement threshold impThreshold=0.01*(fmax-fmin); acqFcn=@(x) expected_improvement_plus(x,model,fmin,impThreshold); else % Use acquisition function specified in options acqFcn=str2func(opts.AcquisitionFunctionName); end xnext=bayesopt_acq(acqFcn,model,lb,ub,opts.AcquisitionSamples); % Evaluate function at next point ynext=f(xnext); % Update data X=[X; xnext(:)]; Y=[Y; ynext]; if ynext < Ybest Xbest=xnext; Ybest=ynext; end fmin=min(Y); fmax=max(Y); % Check stopping criterion if i >=opts.MaxIterations || (i > 1 && abs(Y(end)-Y(end-1))/Ybest <=opts.TolFun) break; end end % Return best point found xopt=Xbest; fopt=Ybest; end function EI=expected_improvement_plus(X,model,fmin,impThreshold) % EXPECTED_IMPROVEMENT_PLUS: Expected improvement with small positive improvement threshold % EI=EXPECTED_IMPROVEMENT_PLUS(X,MODEL,FMIN,IMPTHRESHOLD) computes % the expected improvement (EI) of a surrogate model at the point X. % The input MODEL is a regression model, FMIN is the current minimum % value of the function being modeled, and IMPTHRESHOLD is a small % positive improvement threshold. % % The expected improvement is defined as: % EI=E[max(FMIN - Y, 0)] % where Y is the predicted value of the surrogate model at X. % The expected value is taken over the posterior distribution of Y. % % However, if the predicted value Y is within IMPTHRESHOLD of FMIN, % then EI is set to IMPTHRESHOLD instead. This is done to encourage % exploration of the search space, even if the expected improvement % is very small. % % See also BAYESOPT_ACQ. % Check inputs narginchk(4,4); % Compute predicted value and variance at X [Y,~,sigma]=predict(model,X); % Compute expected improvement z=(fmin - Y - impThreshold)/sigma; EI=(fmin - Y - impThreshold)*normcdf(z) + sigma*normpdf(z); EI(sigma==0)=0; % avoid division by zero % Check if improvement is small if Y >=fmin - impThreshold EI=impThreshold; end end function opts=bayesopt_options(varargin) % BAYESOPT_OPTIONS: Create options structure for Bayesian optimization % OPTS=BAYESOPT_OPTIONS() creates an options structure with default % values for all parameters. % % OPTS=BAYESOPT_OPTIONS(P1,V1,P2,V2,...) creates an options structure % with parameter names and values specified in pairs. Any unspecified % parameters will take on their default values. % % OPTS=BAYESOPT_OPTIONS(OLDOPTS,P1,V1,P2,V2,...) creates a copy of % the OLDOPTS structure, with any parameters specified in pairs % overwriting the corresponding values. % % Available parameters: % MaxIterations - Maximum number of iterations (default 100) % TolFun - Tolerance on function value improvement (default 1e-6) % KernelFunction - Name of kernel function for Gaussian process % regression (default 'squaredexponential') % KernelParameters - Parameters of kernel function (default []) % AcquisitionFunctionName - Name of acquisition function for deciding % which point to evaluate next (default % 'expected-improvement-plus') % AcquisitionSamples - Number of samples to use when evaluating the % acquisition function (default 1000) % % See also BAYESOPT_FUN, BAYESOPT_ACQ. % Define default options opts=struct('MaxIterations',100,'TolFun',1e-6, ... 'KernelFunction','squaredexponential','KernelParameters',[], ... 'AcquisitionFunctionName','expected-improvement-plus','AcquisitionSamples',1000); % Overwrite default options with user-specified options if nargin > 0 if isstruct(varargin{1}) % Copy old options structure and overwrite fields with new values oldopts=varargin{1}; for i=2:2:nargin fieldname=validatestring(varargin{i},fieldnames(opts)); oldopts.(fieldname)=varargin{i+1}; end opts=oldopts; else % Overwrite fields of default options with new values for i=1:2:nargin fieldname=validatestring(varargin{i},fieldnames(opts)); opts.(fieldname)=varargin{i+1}; end end end end function xnext=bayesopt_acq(acqFcn,model,lb,ub,nSamples) % BAYESOPT_ACQ: Find next point to evaluate using an acquisition function % XNEXT=BAYESOPT_ACQ(ACQFCN,MODEL,LB,UB,NSAMPLES) finds the next point % to evaluate using the acquisition function ACQFCN and the regression % model MODEL. LB and UB are the lower and upper bounds of the variables, % and NSAMPLES is the number of random samples to use when maximizing % the acquisition function. % % The input ACQFCN should be a function handle that takes a regression % model and a set of input points as inputs, and returns a vector of % acquisition function values. The set of input points is a matrix with % one row per point and one column per variable. % % The output XNEXT is a vector containing the next point to evaluate. % % See also BAYESOPT_FUN, EXPECTED_IMPROVEMENT_PLUS. % Check inputs narginchk(4,5); assert(isa(acqFcn,'function_handle'),'ACQFCN must be a function handle'); assert(isa(model,'RegressionGP'),'MODEL must be a regressionGP object'); assert(isvector(lb) && isnumeric(lb),'LB must be a numeric vector'); assert(isvector(ub) && isnumeric(ub),'UB must be a numeric vector'); assert(all(size(lb)==size(ub)),'LB and UB must have the same size'); if nargin < 5, nSamples=1000; end % Generate random samples X=bsxfun(@plus,lb,bsxfun(@times,rand(nSamples,numel(lb)),ub-lb)); % Evaluate acquisition function acq=acqFcn(model,X); % Find maximum of acquisition function [~,imax]=max(acq); xnext=X(imax,:); end ``` 该示例代码实现了一个使用贝叶斯优化的函数优化器。该优化器使用高斯过程回归模型来近似目标函数，并使用期望改进加上（EI+）作为获取函数。您可以将此代码用作自己的优化问题的起点，并根据需要进行修改。