adding more to codes for sim

doc/src/SupportVMachines/svm.do.txt

@@ -583,6 +583,27 @@ from which we also find $b$.
 !split 
 ===== Mercer's theorem =====
 
+!split
+=====  Mathematical optimization of convex functions =====
+
+A mathematical optimization problem, or just optimization problem, has the form
+!bt
+\[
+\mathrm{minimize}\hspace{0.1cm} f(x),
+\]
+!et
+subject to some constraints $g(\lambda_i) \leq  b_i$ for say a selected set $i=1,2,\dots, n$.
+In our case we are optimizing with respect to the Lagrangian multipliers $\lambda_i$, and the
+vector $\bm{\lambda}=[\lambda_1, \lambda_2,\dots, \lambda_n]$ is the optimization variable we are dealing with.
+and $f(x)$ is our objective function while $g(\lambda_i) \leq  b_i$ represents our constraint function.
+
+In our case we are particularly interested in a class of optimization problems called convex optmization problems. 
+In our disussion on gradient descent methods we discussed at length the definition of a convex function. 
+
+Convex optimization problems play a central role in applied mathematics and we recommend strongly "Boyd and Vandenberghe's text on the topics":"http://web.stanford.edu/~boyd/cvxbook/".
+
+
+
 !split
 ===== How do we solve these problems =====
 
@@ -625,8 +646,23 @@ object; all arguments given to its solvers must be in this matrix type. There ar
 to do this. The first is to define the matrix directly with (potentially nested) lists:
 from cvxopt import matrix
 !bc pycod
+# Import the necessary packages
+import numpy
+from cvxopt import matrix
+from cvxopt import solvers
+# Define QP parameters (directly)
 P = matrix([[1.0,0.0],[0.0,0.0]])
 q = matrix([3.0,4.0])
 G = matrix([[-1.0,0.0,-1.0,2.0,3.0],[0.0,-1.0,-3.0,5.0,4.0]])
 h = matrix([0.0,0.0,-15.0,100.0,80.0])
+# Define QP parameters (with NumPy)
+P = matrix(numpy.diag([1,0]), tc=’d’)
+q = matrix(numpy.array([3,4]), tc=’d’)
+G = matrix(numpy.array([[-1,0],[0,-1],[-1,-3],[2,5],[3,4]]), tc=’d’)
+h = matrix(numpy.array([0,0,-15,100,80]), tc=’d’)
+# Construct the QP, invoke solver
+sol = solvers.qp(P,q,G,h)
+# Extract optimal value and solution
+sol[’x’] # [7.13e-07, 5.00e+00]
+sol[’primal objective’]
 !ec