L1Solve: Difference between revisions
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<math> | <math> | ||
\begin{array}{cccccc} | \begin{array}{cccccc} | ||
\min\limits_x & | \min\limits_x & {\sum_i |r_i(x)|} \\\mbox{subject to} & x_L & \leq & x & \leq & x_U \\{} & b_L & \leq & Ax & \leq & b_U \\{} & c_L & \leq & c(x) & \leq & c_U \\ | ||
\end{array} | \end{array} | ||
</math> | </math> | ||
where <math>x,x_L,x_U \in \ | where <math>x,x_L,x_U \in \mathbb{R}^{n}</math> , <math>r(x) \in \mathbb{R}^{N}</math> , <math>c(x),c_L,c_U\in \mathbb{R}^{m_1}</math> , <math>b_L,b_U \in \mathbb{R}^{m_2}</math> and <math>A\in \mathbb{R}^{m_2 \times n}</math>. | ||
==Calling Syntax== | ==Calling Syntax== | ||
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<math> | <math> | ||
\begin{array}{cccccc} | \begin{array}{cccccc} | ||
\min\limits_x & | \min\limits_x & {\sum_i (y_i+z_i) } \\ | ||
\mbox{subject to} & x_L & \leq & x & \leq & x_U \\ | \mbox{subject to} & x_L & \leq & x & \leq & x_U \\ | ||
{} & 0 & \leq & y & \leq & \infty \\ | {} & 0 & \leq & y & \leq & \infty \\ | ||
Line 97: | Line 97: | ||
A problem with ''N ''residuals is extended with 2''N ''nonnegative variables <math>y,z \in \ | A problem with ''N ''residuals is extended with 2''N ''nonnegative variables <math>y,z \in \mathbb{R}^{N}</math> along with ''N'' equality constraints <math>r_i(x) + y_i - z_i = 0</math>. | ||
==See Also== | ==See Also== | ||
[[infSolve|''infSolve'']] | [[infSolve|''infSolve'']] |
Revision as of 08:29, 7 December 2011
Purpose
Find a constrained L1 solution of a function of several variables with the use of any suitable nonlinear TOMLAB solver.
L1Solve solves problems of the type:
where , , , and .
Calling Syntax
Result = L1Solve(Prob,PriLev)
Inputs
Prob | Problem description structure. Prob should be created in the cls constrained nonlinear format. | |
---|---|---|
L1Solve uses one special field in Prob: | ||
SolverL1 | Name of the TOMLAB solver used to solve the augmented general nonlinear problem generated by L1Solve. | |
Any other fields are passed along to the solver specified by Prob.SolverL1. In particular: | ||
A | Linear constraint matrix. | |
b_L | Lower bounds on variables. | |
b_U | Upper bounds on variables. | |
c_L | Lower bounds for nonlinear constraints. | |
c_U | Upper bounds for nonlinear constraints.. | |
x_L | Lower bounds on variables. | |
x_U | Upper bounds on variables. | |
x_0 | Starting point. | |
ConsPattern | Nonzero patterns of constraint and residual Jacobians. | |
JacPattern | Prob.LS.y must have the correct residual length if JacPattern is empty but ConsPattern is not.
L1Solve will create the new patterns for the sub-solver using the information supplied in these two fields. | |
PriLev | Print level in L1Solve. | |
= 0 | silent except for error messages. | |
> 0 | print summary information about problem transformation.
Calls PrintResult with specified PriLev. | |
= 2 | standard output from PrintResult. |
Outputs
Result | Structure with results from optimization.
Fields changed depends on which solver was used for the extended problem. The fields x_k, r_k, J_k, c_k, cJac, x_0, xState, cState, v_k, are transformed back to the format of the original L1 problem. g k is calculated as J_kT · r k. The returned problem structure Result.Prob is the result after L1Solve transformed the problem, i.e. the altered Prob structure. |
Description
L1Solve solves the L1 problem by reformulating it as the general constrained optimization problem
A problem with N residuals is extended with 2N nonnegative variables along with N equality constraints .