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.

1．MATLAB 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 221122()32f x x x x x =++

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.TolX

x =

1.0e-008 *

-0.7591 0.2665

fval =

1.3953e-016

unpfun2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 detected

x =

1.0e-015 *

0.1110 -0.8882

fval =

6.2862e-031

程序的相关知识：

第一种: fminsearch

Find a minimum of an unconstrained multivariable function

where 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: fminunc

Find a minimum of an unconstrained multivariable function

where 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:

123

123min s.t.02272

f x x x x x x =-≤++≤

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.TolCon

Active Constraints:

2

x =

24.0000

12.0000

12.0000

fval =

-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 <= b. x0 can be a scalar, vector, or matrix.

x = fmincon(fun,x0,A,b,Aeq,beq) minimizes fun subject to the linear equalities Aeq*x = beq as well as A*x <= b. Set A=[] and b=[] if no inequalities exist.

x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub) defines a set of lower and upper bounds on the design variables in x, so that the solution is always in the range

lb <= x <= ub. Set Aeq=[] and beq=[] if no equalities exist.

x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon) subjects the minimization to the nonlinear inequalities c(x) or equalities ceq(x) defined in nonlcon. fmincon optimizes such that c(x) <= 0 and ceq(x) = 0. Set lb=[] and/or ub=[] if no bounds exist.

x = fmincon(fun,x0,A,b,Aeq,beq,lb,ub,nonlcon,options) minimizes with the optimization options specified in the structure options. Use optimset to set these options. Set nonlcon = [] if there are no nonlinear inequality or equality constraints.

[x,fval] = fmincon(...) returns the value of the objective function fun at the solution x.

[x,fval,exitflag] = fmincon(...) returns a value exitflag that describes the exit condition of fmincon.

[x,fval,exitflag,output] = fmincon(...) returns a structure output with information about the optimization.

[x,fval,exitflag,output,lambda] = fmincon(...) returns a structure lambda whose fields contain the Lagrange multipliers at the solution x.

[x,fval,exitflag,output,lambda,grad] = fmincon(...) returns the value of the gradient of fun at the solution x.

[x,fval,exitflag,output,lambda,grad,hessian] = fmincon(...) returns the value of the Hessian at the solution x. See Hessian.

Avoiding Global Variables via Anonymous and Nested Functions explains how to parameterize the objective function fun, if necessary.

1.3 二次规划

程序名: qprogram

qprogram 实例:

22121

1212min ()6s.t.420

,0

f x x x x x x x x =++--≤≥

在命令窗口输入以下信息:： the program is with the quadratic programming

Please input the constraints number of the programming m=4

m =

4

Please input the variant number of the programming n=4

n =

4

Please input cost matrix of the objective function H(n,n)=[2,0;0,2]

H =

2 0

0 2

Please input cost array of the objective function c(n)_T=[6;0]

c =

6

Please input the coefficient matrix of the constraints A(m,n)=[-2,-1;-1,0;0,-1]

A =

-2 -1

-1 0

0 -1

Please input the resource array of the program b(m)_T=[-4,0,0]'

b =

-4

命令窗口输出信息:

Optimization terminated successfully.

The solution of the quadratic is:

x =

1.0000

2.0000

程序的相关知识:

Solve the quadratic programming problem

where H, A, and Aeq are matrices, and f, b, beq, lb, ub, and x are vectors.

语法如下:

x = quadprog(H,f,A,b)

x = quadprog(H,f,A,b,Aeq,beq)

x = quadprog(H,f,A,b,Aeq,beq,lb,ub)

x = quadprog(H,f,A,b,Aeq,beq,lb,ub,x0)

x = quadprog(H,f,A,b,Aeq,beq,lb,ub,x0,options)

[x,fval] = quadprog(...)

[x,fval,exitflag] = quadprog(...)

[x,fval,exitflag,output] = quadprog(...)

[x,fval,exitflag,output,lambda] = quadprog(...)

解释:

x = quadprog(H,f,A,b) returns a vector x that minimizes 1/2*x'*H*x + f'*x subject to A*x <= b.

x = quadprog(H,f,A,b,Aeq,beq) solves the preceding problem while additionally satisfying the equality constraints Aeq*x = beq.

x = quadprog(H,f,A,b,Aeq,beq,lb,ub) defines a set of lower and upper bounds on the design variables, x, so that the solution is in the range lb <= x <= ub.

x = quadprog(H,f,A,b,Aeq,beq,lb,ub,x0) sets the starting point to x0.

x = quadprog(H,f,A,b,Aeq,beq,lb,ub,x0,options) minimizes with the optimization options specified in the structure options. Use optimset to set these options.

[x,fval] = quadprog(...) returns the value of the objective function at x:

? fval = 0.5*x'*H*x + f'*x.

[x,fval,exitflag] = quadprog(...) returns a value exitflag that describes the exit condition of quadprog.

[x,fval,exitflag,output] = quadprog(...) returns a structure output that contains information about the optimization.

[x,fval,exitflag,output,lambda] = quadprog(...) returns a structure lambda whose fields contain the Lagrange multipliers at the solution x.

2. LINGO 程序说明

LINGO 也是解决非线性规划问题的有力工具之一。它使用较为方便，类似于线性规划问题，只要直接在模型窗口键入要解决的问题即可。

程序名:nlinp

nlinp 执行实例:

123

123min s.t.2272

f x x x x x x =-≤++≤ 0

在模型命令窗口键入以下内容:

min =-x1*x2*x3;

x1+2*x2+2*x3>=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: 56

Objective value: -3456.000

Variable Value Reduced Cost

X1 24.00000 0.000000

X2 12.00000 -0.1217727E-06

X3 12.00000 -0.1198846E-06

Row Slack or Surplus Dual Price

1 -3456.000 -1.000000

2 72.00000 0.000000

3 0.000000 144.0000

LINGO提供了很强大的解决优化问题的能力, 可以调用其固定的一些函数解决工程中的较为复杂的问题，也可以与其他软件提供接口, 甚至可以与数据库接口解决问题，还可以进行较为高级的优化问题的求解, 例如: Production Management, Logistics, Finance（投资组合）, Queuing , Marketing等较为复杂问题的求解, 功能很强大. LINGO还提供了全局最优和局部最优的选择以及光滑优化和非光滑优化的解决, 而且系统本身附带了很多经典的优化问题的源程序及实例.