📄 geometric.htm
字号:
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<html>
<head>
<meta http-equiv="Content-Language" content="en-us">
<title>YALMIP Example : Geometric programming</title>
<meta http-equiv="Content-Type" content="text/html; charset=windows-1251">
<meta content="Microsoft FrontPage 6.0" name="GENERATOR">
<meta name="ProgId" content="FrontPage.Editor.Document">
<link href="yalmip.css" type="text/css" rel="stylesheet">
<base target="_self">
</head>
<body leftMargin="0" topMargin="0">
<div align="left">
<table border="0" cellpadding="4" cellspacing="3" style="border-collapse: collapse" bordercolor="#000000" width="100%" align="left" height="100%">
<tr>
<td width="100%" align="left" height="100%" valign="top">
<h2>Geometric programming</h2>
<hr noShade SIZE="1">
<p>
<img border="0" src="exclamationmark.jpg" align="left" width="16" height="16">This
example requires <a href="solvers.htm#mosek">MOSEK</a>,
<a href="solvers.htm#GPPOSY">GPPOSY</a> or
<a href="solvers.htm#fmincon">fmincon</a><br>
<br>
Nonlinear terms can be defined also with negative and non-integer powers.
This can be used to define geometric optimization problems.<br>
<img border="0" src="gemoetric.gif" width="144" height="102" hspace="77" vspace="10"></p>
<p>Geometric programming solvers are capable of
solving a sub-class of geometric problems where <b>c<font face="Times New Roman">≥0</font></b>
with the additional constraint <b>t<font face="Times New Roman">≥0, </font>
</b>so called posynomial geometric programming. The following example is
taken from the <a href="solvers.htm#mosek">MOSEK</a> manual. (note,
the positivity constraint on <b><font face="Times New Roman">t </font></b>
will be added automatically)</p>
<table cellPadding="10" width="100%">
<tr>
<td class="xmpcode">
<pre>t1 = sdpvar(1,1);
t2 = sdpvar(1,1);
t3 = sdpvar(1,1);
obj = (40*t1^-1*t2^-0.5*t3^-1)+(20*t1*t3)+(40*t1*t2*t3);
F = set((1/3)*t1^-2*t2^-2+(4/3)*t2^0.5*t3^-1 < 1);
solvesdp(F,obj);</pre>
</td>
</tr>
</table>
<p>If the geometric program violates the posynomial assumption, an error
will be issued.</p>
<table cellPadding="10" width="100%">
<tr>
<td class="xmpcode">
<pre>solvesdp(F + set(t1-t2 < 1),obj)
Warning: Solver not applicable
<font color="#000000"> ans =
yalmiptime: 0.0600
solvertime: 0
info: 'Solver not applicable'
problem: -4</font></pre>
</td>
</tr>
</table>
<p>YALMIP will automatically convert some simple violations of the
posynomial assumptions, such as lower bounds on monomial terms and
maximization of negative monomials. The following small program maximizes
the volume of an open box, under constraints on the floor and wall area,
and constraints on the relation between the height, width and depth
(example from
<a href="readmore.htm#BOYDETAL2">[S. Boyd, S. Kim, L. Vandenberghe, A. Hassibi]</a>
).</p>
<table cellPadding="10" width="100%">
<tr>
<td class="xmpcode">
<pre>sdpvar h w d
Awall = 1;
Afloor = 1;
F = set(0.5 < h/w < 2) + set(0.5 < d/w < 2);
F = F + set(2*(h*w+h*d) < Awall) + set(w*d < Afloor);
solvesdp(F,-(h*w*d))</pre>
</td>
</tr>
</table>
<p>The posynomial geometric programming problem is not convex in its
standard formulation. Hence, if a general nonlinear solver is applied to
the problem, it will typically fail. However, by performing a suitable
logarithmic variable transformation, the problem is rendered convex.
YALMIP has built-in support for performing this variable change, and solve
the problem using the nonlinear solver
<a href="solvers.htm#fmincon">fmincon</a>. To invoke this module in
YALMIP, use the solver
tag <code>'fmincon-geometric'.</code>Note that this feature mainly is intended for the
<a href="solvers.htm#fmincon">fmincon</a> solver in the MathWorks Optimization Toolbox.
It may work in the
<a href="solvers.htm#fmincon">fmincon</a> solver in
<a target="_blank" href="http://tomlab.biz">TOMLAB</a>, but this has not
been tested to any larger extent.</p>
<table cellPadding="10" width="100%">
<tr>
<td class="xmpcode">
<pre>t1 = sdpvar(1,1);
t2 = sdpvar(1,1);
t3 = sdpvar(1,1);
obj = (40*t1^-1*t2^-0.5*t3^-1)+(20*t1*t3)+(40*t1*t2*t3);
F = set((1/3)*t1^-2*t2^-2+(4/3)*t2^0.5*t3^-1 < 1);
solvesdp(F,obj,sdpsettings('solver','fmincon-geometric'));</pre>
</td>
</tr>
</table>
<p>
<img border="0" src="exclamationmark.jpg" align="left" width="16" height="16"> The current
version of YALMIP has a bug that may cause problems if you have convex
quadratic constraints. To avoid this problem, use <code>
sdpsettings('convertconvexquad',0)</code>. To avoid some other known
issues, explicitly tell YALMIP that the
problem is a geometric problem by specifying the solver to <code>'gpposy'</code>, <code>'mosek-geometric'</code>
or <code>'fmincon-geometric'</code>.</p>
<p>
<img border="0" src="exclamationmark.jpg" align="left" width="16" height="16">
Never use the commands <b>sqrt</b> and <b>cpower</b> when working with
geometric programs, i.e. always use the ^ operator. The reason is
implementation issues in YALMIP. The commands <b>sqrt</b> and <b>cpower</b>
are meant to be used in optimization problems where a conic model is
derived using convexity propagation, see <a href="extoperators.htm">
nonlinear operators</a>.</p>
<h3>Generalized geometric programming</h3>
<p>Some geometric programs, although not given in standard form, can still
be solved using a standard geometric programming solver after some some
additional variables and constraints have been introduced. YALMIP has
built-in support for some of these conversion.
</p>
<p>To begin with, nonlinear operators can be used also in geometric
programs, as in any other optimization problems (as long as YALMIP is
capable of proving convexity, see the <a href="extoperators.htm">nonlinear
operator examples</a>)</p>
<table cellPadding="10" width="100%">
<tr>
<td class="xmpcode">
<pre>t1 = sdpvar(1,1);
t2 = sdpvar(1,1);
t3 = sdpvar(1,1);
obj = (40*t1^-1*t2^-0.5*t3^-1)+(20*t1*t3)+(40*t1*t2*t3);
F = set(max((1/3)*t1^-2*t2^-2+(4/3)*t2^0.5*t3^-1,0.25*t1*t2) < min(t1,t2));
solvesdp(F,obj);</pre>
</td>
</tr>
</table>
<p>Powers of posynomials are allowed in generalized geometric
programs. YALMIP will automatically take care of this and convert the
problems to a standard geometric programs. Note that the power has to be
positive if used on the left-hand side of a <, and negative otherwise.</p>
<table cellPadding="10" width="100%">
<tr>
<td class="xmpcode">
<pre>t1 = sdpvar(1,1);
t2 = sdpvar(1,1);
t3 = sdpvar(1,1);
obj = (40*t1^-1*t2^-0.5*t3^-1)+(20*t1*t3)+(40*t1*t2*t3);
F = set(max((1/3)*t1^-2*t2^-2+(4/3)*t2^0.5*t3^-1,0.25*t1*t2) < min((t1+0.5*t2)^-1,t2));
F = F + set((2*t1+3*t2^-1)^0.5 < 2);
solvesdp(F,obj);</pre>
</td>
</tr>
</table>
<p>To understand how a generalized geometric program can be converted to a
standard geometric program. the reader is referred to
<a href="readmore.htm#BOYDETAL2">[S. Boyd, S. Kim, L. Vandenberghe, A. Hassibi]</a>
<h3><a name="migp"></a>Mixed integer geometric programming</h3>
<p>The branch and bound solver in YALMIP is built in a modular fashion that makes it
possible to solve almost arbitrary convex mixed integer programs. The
following example is taken from
<a href="readmore.htm#BOYDETAL2">[S. Boyd, S. Kim, L. Vandenberghe, A. Hassibi]</a>.
To begin with, define the data for the example.<table cellPadding="10" width="100%" id="table2">
<tr>
<td class="xmpcode">
<pre>a = ones(7,1);
alpha = ones(7,1);
beta = ones(7,1);
gamma = ones(7,1);
f = [1 0.8 1 0.7 0.7 0.5 0.5]';
e = [1 2 1 1.5 1.5 1 2]';
Cout6 = 10;
Cout7 = 10;</pre>
</td>
</tr>
</table>
<p>Introduce symbolic expressions used in the model.</p>
<table cellPadding="10" width="100%" id="table3">
<tr>
<td class="xmpcode">
<pre>x = sdpvar(7,1);</pre>
<pre>C = alpha+beta.*x;
A = sum(a.*x);
P = sum(f.*e.*x);
R = gamma./x;</pre>
<pre>D1 = R(1)*(C(4));
D2 = R(2)*(C(4)+C(5));
D3 = R(3)*(C(5)+C(7));
D4 = R(4)*(C(6)+C(7));
D5 = R(5)*(C(7));
D6 = R(6)*Cout6;
D7 = R(7)*Cout7;</pre>
</td>
</tr>
</table>
<p>The objective function and constraints (notice the use of the
<a href="extoperators.htm">nonlinear operator</a> <b>max</b> in the
objective).</p>
<table cellPadding="10" width="100%" id="table4">
<tr>
<td class="xmpcode">
<pre>% Constraints
F = set(x > 1) + set(P < 20) + set(A < 100);</pre>
<pre>% Objective
D = max((D1+D4+D6),(D1+D4+D7),(D2+D4+D6),(D2+D4+D7),(D2+D5+D7),(D3+D5+D6),(D3+D7));</pre>
</td>
</tr>
</table>
<p>Solve!</p>
<table cellPadding="10" width="100%" id="table5">
<tr>
<td class="xmpcode">
<pre>solvesdp(F+set(integer(x)),D)
double(D)
<font color="#000000">ans =
8.3333</font></pre>
</td>
</tr>
</table>
<p>An alternative model is discussed in the paper, and is just as easy
to define.</p>
<table cellPadding="10" width="100%" id="table1">
<tr>
<td class="xmpcode">
<pre>T1 = D1;
T2 = D2;
T3 = D3;
T4 = max(T1,T2)+D4;
T5 = max(T2,T3) + D5;
T6 = T4 + D6;
T7 = max(T3,T4,T5) + D7;
D = max(T6,T7);
solvesdp(F+set(integer(x)),D)
double(D)
<font color="#000000">ans =
8.3333</font></pre>
</td>
</tr>
</table>
</td>
</tr>
</table>
<p> </div>
</body>
</html>⌨️ 快捷键说明
复制代码
Ctrl + C
搜索代码
Ctrl + F
全屏模式
F11
切换主题
Ctrl + Shift + D
显示快捷键
?
增大字号
Ctrl + =
减小字号
Ctrl + -