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<title>YALMIP Example : Moment relaxations</title>
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<h2>Moment based relaxation of polynomials programs</h2>
<hr noShade SIZE="1">
<p>YALMIP comes with a built-in module for polynomial programming using
moment relaxations. This can be used for finding lower bounds on constrained
polynomial programs (inequalities and equalities, element-wise and semidefinite), and to extract the corresponding optimizers. The implementation is entirely based on high-level
YALMIP code, and can be somewhat inefficient for large problems (the
inefficiency would then show in the setup of the problem, not actually
solving the semidefinite resulting program). For very large problems, you might
want to check out the dedicated software package
<a target="_blank" href="http://www.laas.fr/~henrion/software/gloptipoly/">Gloptipoly</a>
(the solution time will be the same, but the setup time might be reduced).
For the underlying theory of moment relaxations, the reader is referred to
<a href="readmore.htm#LASSERRE">[Lasserre]</a>.</p>
<h3>Solving polynomial problems by relaxations</h3>
<p>The following code calculates a lower bound on a concave quadratic
optimization problem. As you can see, the only difference compared to
solving the problem using a standard solver, such as
<a href="solvers.htm#fmincon">fmincon</a> or <a href="solvers.htm#penbmi">
penbmi</a>, is that we call <a href="reference.htm#solvemoment">solvemoment</a> instead of
<a href="reference.htm#solvesdp">solvesdp</a>.</p>
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<pre>sdpvar x1 x2 x3
obj = -2*x1+x2-x3;
F = set(x1*(4*x1-4*x2+4*x3-20)+x2*(2*x2-2*x3+9)+x3*(2*x3-13)+24>0);
F = F + set(4-(x1+x2+x3)>0);
F = F + set(6-(3*x2+x3)>0);
F = F + set(x1>0);
F = F + set(2-x1>0);
F = F + set(x2>0);
F = F + set(x3>0);
F = F + set(3-x3>0);
solvemoment(F,obj);
double(obj)
<font color="#000000">
ans =</font></pre>
<pre><font color="#000000"> -6.0000</font></pre>
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<p>Notice that YALMIP does not recover variables by default, a fact showing up in the
difference between lifted variables and actual nonlinear variables (lifted
variables are the variables used in the semidefinite relaxation to model
nonlinear variables) The lifted variables can be obtained by using the
command <code>relaxdouble</code> . The quadratic
constraint above is satisfied in the lifted variables, but not in the true
variables, as the following code illustrates.</p>
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<pre>relaxdouble(x1*(4*x1-4*x2+4*x3-20)+x2*(2*x2-2*x3+9)+x3*(2*x3-13)+24)
<font color="#000000">ans =
23.2648</font></pre>
<pre>double(x1*(4*x1-4*x2+4*x3-20)+x2*(2*x2-2*x3+9)+x3*(2*x3-13)+24)
<font color="#000000"> ans =
-2.0000</font></pre>
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<p>An tighter relaxation can be obtained by using a higher order relaxation (the
lowest possible is used if it is not specified).</p>
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<pre>solvemoment(F,obj,[],2);
double(obj)
<font color="#000000"> ans =</font></pre>
<pre><font color="#000000"> -5.6593</font></pre>
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<p>The obtained bound can be used iteratively to improve the bound by adding
dynamically generated cuts.</p>
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<pre>solvemoment(F+set(obj>double(obj)),obj,[],2);
double(obj)
<font color="#000000">ans =</font></pre>
<pre><font color="#000000"> -5.3870</font></pre>
<pre>solvemoment(F+set(obj>double(obj)),obj,[],2);
double(obj)
<font color="#000000">ans =
-5.1270</font></pre>
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<p>The known true minima, -4, is found in the fourth order relaxation.</p>
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<pre>solvemoment(F,obj,[],4);
double(obj)
<font color="#000000"> ans =</font></pre>
<pre><font color="#000000"> -4.0000</font></pre>
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<p>The true global minima is however not recovered with the lifted variables, as we can see if we
check the current solution (still violates the nonlinear constraint).</p>
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<pre>checkset(F)
<font color="#000000">
+++++++++++++++++++++++++++++++++++++++++++++++++++
| ID| Constraint | Type| Primal residual|
+++++++++++++++++++++++++++++++++++++++++++++++++++
| #1| Numeric value| Element-wise| </font><font color="#FF0000">-0.88573</font><font color="#000000">|
| #2| Numeric value| Element-wise| 1.834|
| #3| Numeric value| Element-wise| 5.668|
| #4| Numeric value| Element-wise| 1.834|
| #5| Numeric value| Element-wise| 0.16599|
| #6| Numeric value| Element-wise| 2.0873e-006|
| #7| Numeric value| Element-wise| 0.33198|
| #8| Numeric value| Element-wise| 2.668|
+++++++++++++++++++++++++++++++++++++++++++++++++++</font></pre>
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<h3>Extracting solutions</h3>
<p>To extract a (or several) globally optimal solution, we need two output
arguments. The first output is a diagnostic structure (standard solution
structure from the semidefinite solver), the second output is the
(hopefully) extracted globally optimal solutions and the third output is
a data structure containing all data that was needed to extract the
solution.</p>
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<pre>[sol,x] = solvemoment(F,obj,[],4);
x{1}
<font color="#000000">
ans =
0.5000
0.0000
3.0000</font>
x{2}
<font color="#000000">
ans =
2.0000
-0.0000
0.0000</font></pre>
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<p>We can easily check that these are globally optimal solutions since
they reach the lower bound -4 and satisfy the constraints (up to
numerical precision).</p>
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<pre>assign([x1;x2;x3],x{1});
double(obj)
<font color="#000000">ans =</font></pre>
<pre><font color="#000000"> -4.0000</font></pre>
<pre>checkset(F)
<font color="#000000">+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
| ID| Constraint| Type| Primal residual|
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
| #1| Numeric value| Element-wise (quadratic)| -1.1034e-006|
| #2| Numeric value| Element-wise| 0.5|
| #3| Numeric value| Element-wise| 3|
| #4| Numeric value| Element-wise| 0.5|
| #5| Numeric value| Element-wise| 1.5|
| #6| Numeric value| Element-wise| 5.956e-007|
| #7| Numeric value| Element-wise| 3|
| #8| Numeric value| Element-wise| 4.6084e-007|
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++</font></pre>
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<h3>Polynomial semidefinite constraints</h3>
<p>Nonlinear semidefinite constraints can be
added using exactly the same notation and syntax. The following example is taken from
<a href="readmore.htm#HENRIONLASSERRE">[D. Henrion, J. B. Lasserre]</a>.</p>
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<pre>sdpvar x1 x2
obj = -x1^2-x2^2;
F = set([1-4*x1*x2 x1;x1 4-x1^2-x2^2]);
[sol,x] = solvemoment(F,obj,[],2);
assign([x1;x2],x{1});
double(obj)
<font color="#000000">ans =</font></pre>
<pre><font color="#000000"> -4.00003</font></pre>
<pre>checkset(F)
<font color="#000000">++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
| ID| Constraint| Type| Primal residual|
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
| #1| Numeric value| LMI (quadratic)| -0.00034633|
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++</font></pre>
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<h3>Advanced features</h3>
<p>A number of advanced features are available. We just briefly introduce
these here by a quick example where we refine the extracted solution
using a couple of Newton steps on an algebraic systems defining the global
solutions given the optimal moment matrices, and change the numerical tolerance in the extraction
algorithm. Finally, we calculate some different global solutions using
the optimal moment matrices. Please check the moment relaxation example in
<a href="reference.htm#yalmipdemo">yalmipdemo</a> for details.</p>
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<pre>sdpvar x1 x2
obj = -x1^2-x2^2;
F = set([1-4*x1*x2 x1;x1 4-x1^2-x2^2]);
ops = sdpsettings('moment.refine',5','moment.rceftol',1e-8);
[sol,xe,data] = solvemoment(F,obj,ops);
xopt1 = extractsolution(data,sdpsettings('moment.refine',0));
xopt2 = extractsolution(data,sdpsettings('moment.refine',1));
xopt3 = extractsolution(data,sdpsettings('moment.refine',10));
xopt4 = extractsolution(data,sdpsettings('moment.rceftol',1e-3,'moment.refine',5));</pre>
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<p>The moment relaxation solver can also be called using a more standard
YALMIP notation, by simply defining the solver as <code>'moment'</code>.</p>
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<pre>sdpvar x1 x2
obj = -x1^2-x2^2;
F = set([1-4*x1*x2 x1;x1 4-x1^2-x2^2]);
sol = solvesdp(F,obj,sdpsettings('solver','moment','moment.order',2));
assign(sol.momentdata.x,sol.xoptimal{1});
double(obj)
<font color="#000000">ans =</font></pre>
<pre><font color="#000000"> -4.00003</font></pre>
<pre>checkset(F)
<font color="#000000">++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
| ID| Constraint| Type| Primal residual|
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
| #1| Numeric value| LMI (quadratic)| -0.00034633|
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++</font></pre>
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<p>
<img border="0" src="demoicon.gif" width="16" height="16"> The
semidefinite programs rapidly grows when the number of variables and the
polynomial degree increase, so be careful when you model your problem.</p>
<p>
<img border="0" src="demoicon.gif" width="16" height="16"> The two
most useful practical tips when working semidefinite relaxations seem to
be to de-symmetrize your objective function, and compactify your
feasible region. These two tricks typically increase the likelihood that
you will be able to extract global solutions. By adding a perturbation
to the polynomial, symmetry is lost, which generically means that there
will not be infinitely many optima, and the extraction algorithm is more
likely to work. Most theory in moment relaxations assumes that the
feasible set is compact, and this is also affecting practical
performance. By adding redundant compactifying constraints, you
typically increase the likelihood of success. As an example, a simple
redundant constraint which often work well in practice is an upper bound on
the objective function based on a known feasible sub-optimal solution.</td>
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<p> </div>
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