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<h2>Geometric programming</h2>
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<p>
<img border="0" src="exclamationmark.jpg" align="left" width="16" height="16">This
example requires <a href="solvers.htm#mosek">MOSEK</a> (or
<a href="solvers.htm#fmincon">fmincon</a>, see below). <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>The solver <a href="solvers.htm#mosek">MOSEK</a> is 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>
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<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>
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<p>If the geometric program violates the posynomial assumption, an error
will be issued.</p>
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<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>
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<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>
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<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>
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<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 fmincon. 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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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