1. 简述
1.fminsearch函数基本语法
函数功能：使用无导数法计算无约束多变量函数的最小值
语法
x = fminsearch(fun,x0)
x = fminsearch(fun,x0,options)
x = fminsearch(problem)
[x,fval] = fminsearch(___)
[x,fval,exitflag] = fminsearch(___)
[x,fval,exitflag,output] = fminsearch(___)
说明
非线性规划求解器。搜索由以下公式指定的问题的最小值：
min f(x)
f(x) 是返回标量的函数，x 是向量或矩阵；请参阅矩阵参数。
x = fminsearch(fun,x0) 在点 x0 处开始并尝试求 fun 中描述的函数的局部最小值 x。
x = fminsearch(fun,x0,options) 使用 options 所指定的优化选项执行最小化。使用 optimset 可设置这些选项。
x = fminsearch(problem) 求 problem 的最小值，它是 problem 中所述的一个结构体。
[x,fval] = fminsearch(___)，对任何上述输入语法，在 fval 中返回目标函数 fun 在解 x 处的值。
[x,fval,exitflag] = fminsearch(___) 还返回描述退出条件的值 exitflag。
[x,fval,exitflag,output] = fminsearch(___) 还会返回结构体 output 以及有关优化过程的信息。2. 代码
主程序：
% 通过绘图确定一个初始值；然后进行迭代找到真正的最小值；
clc
clear
x,y\]=meshgrid(-6:.5:6); f= 8\*x-4\*y +x.\^2+3\*y.\^2; surfc(x,y,f) x0=\[0,0\]; %\[x,fval,exitflag\]=fminunc(@(x)(8\*x(1)-4\*x(2) +x(1).\^2+3\*x(2).\^2),x0) options=optimset('display','iter','Tolx',1e-8); \[x,fval,exitflag\]=fminunc(@(x)(8\*x(1)-4\*x(2) +x(1).\^2+3\*x(2).\^2),x0,options) \[x_fminsearch,fval_fminsearch,exitflag\]=fminsearch(@(x)(8\*x(1)-4\*x(2) +x(1).\^2+3\*x(2).\^2),x0,options) 子程序： function \[x,fval,exitflag,output\] = fminsearch(funfcn,x,options,varargin) %FMINSEARCH Multidimensional unconstrained nonlinear minimization (Nelder-Mead). % X = FMINSEARCH(FUN,X0) starts at X0 and attempts to find a local minimizer % X of the function FUN. FUN is a function handle. FUN accepts input X and % returns a scalar function value F evaluated at X. X0 can be a scalar, vector % or matrix. % % X = FMINSEARCH(FUN,X0,OPTIONS) minimizes with the default optimization % parameters replaced by values in the structure OPTIONS, created % with the OPTIMSET function. See OPTIMSET for details. FMINSEARCH uses % these options: Display, TolX, TolFun, MaxFunEvals, MaxIter, FunValCheck, % PlotFcns, and OutputFcn. % % X = FMINSEARCH(PROBLEM) finds the minimum for PROBLEM. PROBLEM is a % structure with the function FUN in PROBLEM.objective, the start point % in PROBLEM.x0, the options structure in PROBLEM.options, and solver % name 'fminsearch' in PROBLEM.solver. % % \[X,FVAL\]= FMINSEARCH(...) returns the value of the objective function, % described in FUN, at X. % % \[X,FVAL,EXITFLAG\] = FMINSEARCH(...) returns an EXITFLAG that describes % the exit condition. Possible values of EXITFLAG and the corresponding % exit conditions are % % 1 Maximum coordinate difference between current best point and other % points in simplex is less than or equal to TolX, and corresponding % difference in function values is less than or equal to TolFun. % 0 Maximum number of function evaluations or iterations reached. % -1 Algorithm terminated by the output function. % % \[X,FVAL,EXITFLAG,OUTPUT\] = FMINSEARCH(...) returns a structure % OUTPUT with the number of iterations taken in OUTPUT.iterations, the % number of function evaluations in OUTPUT.funcCount, the algorithm name % in OUTPUT.algorithm, and the exit message in OUTPUT.message. % % Examples % FUN can be specified using @: % X = fminsearch(@sin,3) % finds a minimum of the SIN function near 3. % In this case, SIN is a function that returns a scalar function value % SIN evaluated at X. % % FUN can be an anonymous function: % X = fminsearch(@(x) norm(x),\[1;2;3\]) % returns a point near the minimizer \[0;0;0\]. % % FUN can be a parameterized function. Use an anonymous function to % capture the problem-dependent parameters: % f = @(x,c) x(1).\^2+c.\*x(2).\^2; % The parameterized function. % c = 1.5; % The parameter. % X = fminsearch(@(x) f(x,c),\[0.3;1\]) % % FMINSEARCH uses the Nelder-Mead simplex (direct search) method. % % See also OPTIMSET, FMINBND, FUNCTION_HANDLE. % Reference: Jeffrey C. Lagarias, James A. Reeds, Margaret H. Wright, % Paul E. Wright, "Convergence Properties of the Nelder-Mead Simplex % Method in Low Dimensions", SIAM Journal of Optimization, 9(1): % p.112-147, 1998. % Copyright 1984-2018 The MathWorks, Inc. defaultopt = struct('Display','notify','MaxIter','200\*numberOfVariables',... 'MaxFunEvals','200\*numberOfVariables','TolX',1e-4,'TolFun',1e-4, ... 'FunValCheck','off','OutputFcn',\[\],'PlotFcns',\[\]); % If just 'defaults' passed in, return the default options in X if nargin == 1 \&\& nargout \<= 1 \&\& strcmpi(funfcn,'defaults') x = defaultopt; return end if nargin \< 3, options = \[\]; end % Detect problem structure input if nargin == 1 if isa(funfcn,'struct') \[funfcn,x,options\] = separateOptimStruct(funfcn); else % Single input and non-structure error('MATLAB:fminsearch:InputArg',... getString(message('MATLAB:optimfun:fminsearch:InputArg'))); end end if nargin == 0 error('MATLAB:fminsearch:NotEnoughInputs',... getString(message('MATLAB:optimfun:fminsearch:NotEnoughInputs'))); end % Check for non-double inputs if \~isa(x,'double') error('MATLAB:fminsearch:NonDoubleInput',... getString(message('MATLAB:optimfun:fminsearch:NonDoubleInput'))); end n = numel(x); numberOfVariables = n; % Check that options is a struct if \~isempty(options) \&\& \~isa(options,'struct') error('MATLAB:fminsearch:ArgNotStruct',... getString(message('MATLAB:optimfun:commonMessages:ArgNotStruct', 3))); end printtype = optimget(options,'Display',defaultopt,'fast'); tolx = optimget(options,'TolX',defaultopt,'fast'); tolf = optimget(options,'TolFun',defaultopt,'fast'); maxfun = optimget(options,'MaxFunEvals',defaultopt,'fast'); maxiter = optimget(options,'MaxIter',defaultopt,'fast'); funValCheck = strcmp(optimget(options,'FunValCheck',defaultopt,'fast'),'on'); % In case the defaults were gathered from calling: optimset('fminsearch'): if ischar(maxfun) \|\| isstring(maxfun) if strcmpi(maxfun,'200\*numberofvariables') maxfun = 200\*numberOfVariables; else error('MATLAB:fminsearch:OptMaxFunEvalsNotInteger',... getString(message('MATLAB:optimfun:fminsearch:OptMaxFunEvalsNotInteger'))); end end if ischar(maxiter) \|\| isstring(maxiter) if strcmpi(maxiter,'200\*numberofvariables') maxiter = 200\*numberOfVariables; else error('MATLAB:fminsearch:OptMaxIterNotInteger',... getString(message('MATLAB:optimfun:fminsearch:OptMaxIterNotInteger'))); end end switch printtype case {'notify','notify-detailed'} prnt = 1; case {'none','off'} prnt = 0; case {'iter','iter-detailed'} prnt = 3; case {'final','final-detailed'} prnt = 2; case 'simplex' prnt = 4; otherwise prnt = 1; end % Handle the output outputfcn = optimget(options,'OutputFcn',defaultopt,'fast'); if isempty(outputfcn) haveoutputfcn = false; else haveoutputfcn = true; xOutputfcn = x; % Last x passed to outputfcn; has the input x's shape % Parse OutputFcn which is needed to support cell array syntax for OutputFcn. outputfcn = createCellArrayOfFunctions(outputfcn,'OutputFcn'); end % Handle the plot plotfcns = optimget(options,'PlotFcns',defaultopt,'fast'); if isempty(plotfcns) haveplotfcn = false; else haveplotfcn = true; xOutputfcn = x; % Last x passed to plotfcns; has the input x's shape % Parse PlotFcns which is needed to support cell array syntax for PlotFcns. plotfcns = createCellArrayOfFunctions(plotfcns,'PlotFcns'); end header = ' Iteration Func-count min f(x) Procedure'; % Convert to function handle as needed. funfcn = fcnchk(funfcn,length(varargin)); % Add a wrapper function to check for Inf/NaN/complex values if funValCheck % Add a wrapper function, CHECKFUN, to check for NaN/complex values without % having to change the calls that look like this: % f = funfcn(x,varargin{:}); % x is the first argument to CHECKFUN, then the user's function, % then the elements of varargin. To accomplish this we need to add the % user's function to the beginning of varargin, and change funfcn to be % CHECKFUN. varargin = \[{funfcn}, varargin\]; funfcn = @checkfun; end n = numel(x); % Initialize parameters rho = 1; chi = 2; psi = 0.5; sigma = 0.5; onesn = ones(1,n); two2np1 = 2:n+1; one2n = 1:n; % Set up a simplex near the initial guess. xin = x(:); % Force xin to be a column vector v = zeros(n,n+1); fv = zeros(1,n+1); v(:,1) = xin; % Place input guess in the simplex! (credit L.Pfeffer at Stanford) x(:) = xin; % Change x to the form expected by funfcn fv(:,1) = funfcn(x,varargin{:}); func_evals = 1; itercount = 0; how = ''; % Initial simplex setup continues later % Initialize the output and plot functions. if haveoutputfcn \|\| haveplotfcn \[xOutputfcn, optimValues, stop\] = callOutputAndPlotFcns(outputfcn,plotfcns,v(:,1),xOutputfcn,'init',itercount, ... func_evals, how, fv(:,1),varargin{:}); if stop \[x,fval,exitflag,output\] = cleanUpInterrupt(xOutputfcn,optimValues); if prnt \> 0 disp(output.message) end return; end end % Print out initial f(x) as 0th iteration if prnt == 3 disp(' ') disp(header) fprintf(' %5.0f %5.0f %12.6g %s\\n', itercount, func_evals, fv(1), how); elseif prnt == 4 formatsave.format = get(0,'format'); formatsave.formatspacing = get(0,'formatspacing'); % reset format when done oc1 = onCleanup(@()set(0,'format',formatsave.format)); oc2 = onCleanup(@()set(0,'formatspacing',formatsave.formatspacing)); format compact format short e disp(' ') disp(how) disp('v = ') disp(v) disp('fv = ') disp(fv) disp('func_evals = ') disp(func_evals) end % OutputFcn and PlotFcns call if haveoutputfcn \|\| haveplotfcn \[xOutputfcn, optimValues, stop\] = callOutputAndPlotFcns(outputfcn,plotfcns,v(:,1),xOutputfcn,'iter',itercount, ... func_evals, how, fv(:,1),varargin{:}); if stop % Stop per user request. \[x,fval,exitflag,output\] = cleanUpInterrupt(xOutputfcn,optimValues); if prnt \> 0 disp(output.message) end return; end end % Continue setting up the initial simplex. % Following improvement suggested by L.Pfeffer at Stanford usual_delta = 0.05; % 5 percent deltas for non-zero terms zero_term_delta = 0.00025; % Even smaller delta for zero elements of x for j = 1:n y = xin; if y(j) \~= 0 y(j) = (1 + usual_delta)\*y(j); else y(j) = zero_term_delta; end v(:,j+1) = y; x(:) = y; f = funfcn(x,varargin{:}); fv(1,j+1) = f; end % sort so v(1,:) has the lowest function value \[fv,j\] = sort(fv); v = v(:,j); how = 'initial simplex'; itercount = itercount + 1; func_evals = n+1; if prnt == 3 fprintf(' %5.0f %5.0f %12.6g %s\\n', itercount, func_evals, fv(1), how) elseif prnt == 4 disp(' ') disp(how) disp('v = ') disp(v) disp('fv = ') disp(fv) disp('func_evals = ') disp(func_evals) end % OutputFcn and PlotFcns call if haveoutputfcn \|\| haveplotfcn \[xOutputfcn, optimValues, stop\] = callOutputAndPlotFcns(outputfcn,plotfcns,v(:,1),xOutputfcn,'iter',itercount, ... func_evals, how, fv(:,1),varargin{:}); if stop % Stop per user request. \[x,fval,exitflag,output\] = cleanUpInterrupt(xOutputfcn,optimValues); if prnt \> 0 disp(output.message) end return; end end exitflag = 1; % Main algorithm: iterate until % (a) the maximum coordinate difference between the current best point and the % other points in the simplex is less than or equal to TolX. Specifically, % until max(\|\|v2-v1\|\|,\|\|v3-v1\|\|,...,\|\|v(n+1)-v1\|\|) \<= TolX, % where \|\|.\|\| is the infinity-norm, and v1 holds the % vertex with the current lowest value; AND % (b) the corresponding difference in function values is less than or equal % to TolFun. (Cannot use OR instead of AND.) % The iteration stops if the maximum number of iterations or function evaluations % are exceeded while func_evals \< maxfun \&\& itercount \< maxiter if max(abs(fv(1)-fv(two2np1))) \<= max(tolf,10\*eps(fv(1))) \&\& ... max(max(abs(v(:,two2np1)-v(:,onesn)))) \<= max(tolx,10\*eps(max(v(:,1)))) break end % Compute the reflection point % xbar = average of the n (NOT n+1) best points xbar = sum(v(:,one2n), 2)/n; xr = (1 + rho)\*xbar - rho\*v(:,end); x(:) = xr; fxr = funfcn(x,varargin{:}); func_evals = func_evals+1; if fxr \< fv(:,1) % Calculate the expansion point xe = (1 + rho\*chi)\*xbar - rho\*chi\*v(:,end); x(:) = xe; fxe = funfcn(x,varargin{:}); func_evals = func_evals+1; if fxe \< fxr v(:,end) = xe; fv(:,end) = fxe; how = 'expand'; else v(:,end) = xr; fv(:,end) = fxr; how = 'reflect'; end else % fv(:,1) \<= fxr if fxr \< fv(:,n) v(:,end) = xr; fv(:,end) = fxr; how = 'reflect'; else % fxr \>= fv(:,n) % Perform contraction if fxr \< fv(:,end) % Perform an outside contraction xc = (1 + psi\*rho)\*xbar - psi\*rho\*v(:,end); x(:) = xc; fxc = funfcn(x,varargin{:}); func_evals = func_evals+1; if fxc \<= fxr v(:,end) = xc; fv(:,end) = fxc; how = 'contract outside'; else % perform a shrink how = 'shrink'; end else % Perform an inside contraction xcc = (1-psi)\*xbar + psi\*v(:,end); x(:) = xcc; fxcc = funfcn(x,varargin{:}); func_evals = func_evals+1; if fxcc \< fv(:,end) v(:,end) = xcc; fv(:,end) = fxcc; how = 'contract inside'; else % perform a shrink how = 'shrink'; end end if strcmp(how,'shrink') for j=two2np1 v(:,j)=v(:,1)+sigma\*(v(:,j) - v(:,1)); x(:) = v(:,j); fv(:,j) = funfcn(x,varargin{:}); end func_evals = func_evals + n; end end end \[fv,j\] = sort(fv); v = v(:,j); itercount = itercount + 1; if prnt == 3 fprintf(' %5.0f %5.0f %12.6g %s\\n', itercount, func_evals, fv(1), how) elseif prnt == 4 disp(' ') disp(how) disp('v = ') disp(v) disp('fv = ') disp(fv) disp('func_evals = ') disp(func_evals) end % OutputFcn and PlotFcns call if haveoutputfcn \|\| haveplotfcn \[xOutputfcn, optimValues, stop\] = callOutputAndPlotFcns(outputfcn,plotfcns,v(:,1),xOutputfcn,'iter',itercount, ... func_evals, how, fv(:,1),varargin{:}); if stop % Stop per user request. \[x,fval,exitflag,output\] = cleanUpInterrupt(xOutputfcn,optimValues); if prnt \> 0 disp(output.message) end return; end end end % while x(:) = v(:,1); fval = fv(:,1); output.iterations = itercount; output.funcCount = func_evals; output.algorithm = 'Nelder-Mead simplex direct search'; % OutputFcn and PlotFcns call if haveoutputfcn \|\| haveplotfcn callOutputAndPlotFcns(outputfcn,plotfcns,x,xOutputfcn,'done',itercount, func_evals, how, fval, varargin{:}); end if func_evals \>= maxfun msg = getString(message('MATLAB:optimfun:fminsearch:ExitingMaxFunctionEvals', sprintf('%f',fval))); if prnt \> 0 disp(' ') disp(msg) end exitflag = 0; elseif itercount \>= maxiter msg = getString(message('MATLAB:optimfun:fminsearch:ExitingMaxIterations', sprintf('%f',fval))); if prnt \> 0 disp(' ') disp(msg) end exitflag = 0; else msg = ... getString(message('MATLAB:optimfun:fminsearch:OptimizationTerminatedXSatisfiesCriteria', ... sprintf('%e',tolx), sprintf('%e',tolf))); if prnt \> 1 disp(' ') disp(msg) end exitflag = 1; end output.message = msg; %-------------------------------------------------------------------------- function \[xOutputfcn, optimValues, stop\] = callOutputAndPlotFcns(outputfcn,plotfcns,x,xOutputfcn,state,iter,... numf,how,f,varargin) % CALLOUTPUTANDPLOTFCNS assigns values to the struct OptimValues and then calls the % outputfcn/plotfcns. % % state - can have the values 'init','iter', or 'done'. % For the 'done' state we do not check the value of 'stop' because the % optimization is already done. optimValues.iteration = iter; optimValues.funccount = numf; optimValues.fval = f; optimValues.procedure = how; xOutputfcn(:) = x; % Set x to have user expected size stop = false; state = char(state); % Call output functions if \~isempty(outputfcn) switch state case {'iter','init'} stop = callAllOptimOutputFcns(outputfcn,xOutputfcn,optimValues,state,varargin{:}) \|\| stop; case 'done' callAllOptimOutputFcns(outputfcn,xOutputfcn,optimValues,state,varargin{:}); end end % Call plot functions if \~isempty(plotfcns) switch state case {'iter','init'} stop = callAllOptimPlotFcns(plotfcns,xOutputfcn,optimValues,state,varargin{:}) \|\| stop; case 'done' callAllOptimPlotFcns(plotfcns,xOutputfcn,optimValues,state,varargin{:}); end end %-------------------------------------------------------------------------- function \[x,FVAL,EXITFLAG,OUTPUT\] = cleanUpInterrupt(xOutputfcn,optimValues) % CLEANUPINTERRUPT updates or sets all the output arguments of FMINBND when the optimization % is interrupted. % Call plot function driver to finalize the plot function figure window. If % no plot functions have been specified or the plot function figure no % longer exists, this call just returns. callAllOptimPlotFcns('cleanuponstopsignal'); x = xOutputfcn; FVAL = optimValues.fval; EXITFLAG = -1; OUTPUT.iterations = optimValues.iteration; OUTPUT.funcCount = optimValues.funccount; OUTPUT.algorithm = 'Nelder-Mead simplex direct search'; OUTPUT.message = getString(message('MATLAB:optimfun:fminsearch:OptimizationTerminatedPrematurelyByUser')); %-------------------------------------------------------------------------- function f = checkfun(x,userfcn,varargin) % CHECKFUN checks for complex or NaN results from userfcn. f = userfcn(x,varargin{:}); % Note: we do not check for Inf as FMINSEARCH handles it naturally. if isnan(f) error('MATLAB:fminsearch:checkfun:NaNFval',... getString(message('MATLAB:optimfun:fminsearch:checkfun:NaNFval', localChar( userfcn )))); elseif \~isreal(f) error('MATLAB:fminsearch:checkfun:ComplexFval',... getString(message('MATLAB:optimfun:fminsearch:checkfun:ComplexFval', localChar( userfcn )))); end %-------------------------------------------------------------------------- function strfcn = localChar(fcn) % Convert the fcn to a character array for printing if ischar(fcn) strfcn = fcn; elseif isstring(fcn) \|\| isa(fcn,'inline') strfcn = char(fcn); elseif isa(fcn,'function_handle') strfcn = func2str(fcn); else try strfcn = char(fcn); catch strfcn = getString(message('MATLAB:optimfun:fminsearch:NameNotPrintable')); end end **3.** **运行结果**       