\(\newcommand{\W}[1]{ \; #1 \; }\) \(\newcommand{\R}[1]{ {\rm #1} }\) \(\newcommand{\B}[1]{ {\bf #1} }\) \(\newcommand{\D}[2]{ \frac{\partial #1}{\partial #2} }\) \(\newcommand{\DD}[3]{ \frac{\partial^2 #1}{\partial #2 \partial #3} }\) \(\newcommand{\Dpow}[2]{ \frac{\partial^{#1}}{\partial {#2}^{#1}} }\) \(\newcommand{\dpow}[2]{ \frac{ {\rm d}^{#1}}{{\rm d}\, {#2}^{#1}} }\)

min_nso_linear

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Non-Smooth Optimization Using Abs-normal Linear Approximations

Syntax

ok = min_nso_linear (

      level , g , a , epsilon , maxitr , b_in , x_in , x_out

)

Prototype

template <class DblVector, class SizeVector>
bool min_nso_linear(
   size_t           level     ,
   ADFun<double>&   g         ,
   ADFun<double>&   a         ,
   const DblVector& epsilon   ,
   SizeVector       maxitr    ,
   double           b_in      ,
   const DblVector& x_in      ,
   DblVector&       x_out     )

Source

This following is a link to the source code for this example: min_nso_linear.hpp .

Purpose

Given a current that abs-normal representation g , a , for a function \(f(x)\), this routine minimizes \(f(x)\).

DblVector

is a SimpleVector class with elements of type double .

SizeVector

is a SimpleVector class with elements of type size_t .

f

We use the notation f for the original function; see f .

n

We use n to denote the dimension of the domain for f ; i.e., f . Domain () .

m

We use m to denote the dimension of the range for f ; i.e., f . Range () . This must be equal to one.

s

We use s to denote the number absolute terms in f .

level

This value is less that or equal 5. If level == 0 , no tracing of the optimization is printed. If level >= 1 , a trace of each iteration of min_nso_linear is printed. If level >= 2 , a trace of each iteration of the abs_min_linear sub-problem is printed. If level >= 3 , a trace of the lp_box sub-problem is printed. If level >= 4 , a trace of the objective and primal variables \(x\) are printed at each simplex_method iteration. If level == 5 , the simplex tableau is printed at each simplex iteration.

g

This is the function g in the abs-normal representation of f .

a

This is the function a in the abs-normal representation of f .

epsilon

This is a vector with size 2. The value epsilon [0] is convergence criteria in terms of the infinity norm of the difference of x_out between iterations. The value epsilon [1] is convergence criteria in terms of the derivative of \(f(x)\). This derivative is actually the average of the directional derivative in the direction of the sub-problem minimizer.

maxitr

This is a vector with size 3. The value maxitr [0] is the maximum number of min_nso_linear iterations to try before giving up on convergence. The value maxitr [1] is the maximum number of iterations in the abs_min_linear sub-problem. The value maxitr [2] is the maximum number of iterations in the simplex_method sub-problems.

b_in

This the initial bound on the trust region size. To be specific, if \(b\) is the current trust region size, at each iteration affine approximation is minimized with respect to \(\Delta x\) and subject to

\[-b \leq \Delta x_j \leq b\]

for j = 0 , …, n -1 . It must hold that b_in > epsilon [0] .

x_in

This vector x_out has size n . It is the starting point for the optimization procedure; i.e., the min_nso_linear iterations.

x_out

This vector x_out has size n . The input value of its elements does not matter. Upon return, it is the approximate minimizer of the abs-normal approximation for \(f(x)\) over the trust region is \(x = \hat{x} + \Delta x\).

Example

The file min_nso_linear.cpp contains an example and test of min_nso_linear .