第四章非线性规划

第四章 非线性规划本章, 我们介绍两种解决非线性规划问题的软件:第一种: MATLAB 中的 optimization toolbox 中的若干程序;第二种: LINGO 软件.1MATLAB 程序说明1.1 无约束问题程序名: unpfun1 函数, unpfun2 函数unpfun1 实例:Minimize the function 221()3fxx在命令窗口输入以下信息: x0=1,1; % Then call fminunc to find a minimum of unpfun1 near 1,1 x,fval=fminunc(unpfun1,x0)输出以下信息:Optimization terminated successfully:Search direction less than 2*options.TolXx =1.0e-008 *-0.7591 0.2665fval =1.3953e-016unpfun2 实例：将上述的实例用梯度法做在命令窗口输入以下信息: options = optimset(GradObj,on); % To minimize this function with the gradient provided x0 = 1,1; x,fval = fminunc(unpfun2,x0,options)输出以下信息:Optimization terminated successfully:First-order optimality less than OPTIONS.TolFun, and no negative/zero curvature detectedx =1.0e-015 *0.1110 -0.8882fval =6.2862e-031程序的相关知识：第一种: fminsearchFind a minimum of an unconstrained multivariable functionwhere x is a vector and f(x) is a function that returns a scalar.语法如下:x = fminsearch(fun,x0)x = fminsearch(fun,x0,options)x,fval = fminsearch(.)x,fval,exitflag = fminsearch(.)x,fval,exitflag,output = fminsearch(.)解释:fminsearch attempts to find a minimum of a scalar function of several variables, starting at an initial estimate. This is generally referred to as unconstrained nonlinear optimization.x = fminsearch(fun,x0) starts at the point x0 and attempts to find a local minimum x of the function described in fun. fun is a function handle for either an M-file function or an anonymous function. x0 can be a scalar, vector, or matrix.x = fminsearch(fun,x0,options) minimizes with the optimization options specified in the structure options. Use optimset to set these options. x,fval = fminsearch(.) returns in fval the value of the objective function fun at the solution x.x,fval,exitflag = fminsearch(.) returns a value exitflag that describes the exit condition of fminsearch.x,fval,exitflag,output = fminsearch(.) returns a structure output that contains information about the optimization.Avoiding Global Variables via Anonymous and Nested Functions explains how to parameterize the objective function fun, if necessary.第二种: fminuncFind a minimum of an unconstrained multivariable functionwhere x is a vector and f(x) is a function that returns a scalar.语法如下：x = fminunc(fun,x0)x = fminunc(fun,x0,options)x,fval = fminunc(.)x,fval,exitflag = fminunc(.)x,fval,exitflag,output = fminunc(.)x,fval,exitflag,output,grad = fminunc(.)x,fval,exitflag,output,grad,hessian = fminunc(.)解释：fminunc attempts to find a minimum of a scalar function of several variables, starting at an initial estimate. This is generally referred to as unconstrained nonlinear optimization.x = fminunc(fun,x0) starts at the point x0 and attempts to find a local minimum x of the function described in fun. x0 can be a scalar, vector, or matrix. x = fminunc(fun,x0,options) minimizes with the optimization options specified in the structure options. Use optimset to set these options. x,fval = fminunc(.) returns in fval the value of the objective function fun at the solution x.x,fval,exitflag = fminunc(.) returns a value exitflag that describes the exit condition.x,fval,exitflag,output = fminunc(.) returns a structure output that contains information about the optimization.x,fval,exitflag,output,grad = fminunc(.) returns in grad the value of the gradient of fun at the solution x.x,fval,exitflag,output,grad,hessian = fminunc(.) returns in hessian the value of the Hessian of the objective function fun at the solution x. See Hessian.Avoiding Global Variables via Anonymous and Nested Functions explains how to parameterize the objective function fun, if necessary.1.2 有约束的非线性规划程序名: cnpfun 函数cnfun 实例 : 123mins.t072fx 在命令窗口输入以下信息: A=-1,-2,-2;1,2,2; b=0;72; x0 = 10; 10; 10; % Starting guess at the solution x,fval = fmincon(cnpfun,x0,A,b)输出以下信息:Optimization terminated successfully:Magnitude of directional derivative in search direction less than 2*options.TolFun and maximum constraint violation is less than options.TolConActive Constraints:2x =24.000012.000012.0000fval =-3456程序的相关知识:Find a minimum of a constrained nonlinear multivariable function subject to where x, b, beq, lb, and ub are vectors, A and Aeq are matrices, c(x) and ceq(x) are functions that return vectors, and f(x) is a function that returns a scalar. f(x), c(x), and ceq(x) can be nonlinear functions.语法如下：x = fmincon(fun,x0,A,b)x = fmincon(fun,x0,A,b,Aeq,beq)x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub)x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon)x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options)x,fval = fmincon(.)x,fval,exitflag = fmincon(.)x,fval,exitflag,output = fmincon(.)x,fval,exitflag,output,lambda = fmincon(.)x,fval,exitflag,output,lambda,grad = fmincon(.)x,fval,exitflag,output,lambda,grad,hessian = fmincon(.)解释：fmincon attempts to find a constrained minimum of a scalar function of several variables starting at an initial estimate. This is generally referred to as constrained nonlinear optimization or nonlinear programming.x = fmincon(fun,x0,A,b) starts at x0 and attempts to find a minimum x to the function described in fun subject to the linear inequalities A*x =0;x1+2*x2+2*x3=72;按运行按钮在 solution report 窗口得到以下结果:Local optimal solution found at iteration: 56Objective value: -3456.000Variable Value Reduced CostX1 24.00000 0.000000X2 12.00000 -0.1217727E-06X3 12.00000 -0.1198846E-06Row Slack or Surplus Dual Price1 -3456.000 -1.0000002 72.00000 0