barrierPenalty

Logarithmic barrier penalty value for a point with respect to a bounded region

Syntax

p = barrierPenalty(x,xmin,xmax)

p = barrierPenalty(___,maxValue,curvature)

Description

p = barrierPenalty(x,xmin,xmax) calculates the nonnegative (logarithmic barrier) penalty vector p for the point x with respect to the region bounded by xmin and xmax. p has the same dimension as x. This syntax uses the default values of 1 and 0.1 for the maxValue and curvature parameters of the barrier function, respectively.

p = barrierPenalty(___,maxValue,curvature) specifies both the maxValue and curvature parameters of the barrier function. If maxValue is an empty matrix its default value is used. Likewise if curvature is an empty matrix or it is omitted, its default value is used.

Examples

Calculate Logarithmic Barrier Penalty for a Point

This example shows how to use the logarithmic barrierPenalty function to calculate the barrier penalty for a given point, with respect to a bounded region.

Calculate the penalty value for the point 0.1 within the interval [-2,2] using default values for the maximum value and curvature parameters.

barrierPenalty(0.1,-2,2)

ans = 2.5031e-04

Calculate the penalty value for the point 4 outside the interval [-2,2].

barrierPenalty(4,-2,2)

ans = 1

Calculate the penalty value for the point 4 outside the interval [-2,2], using a maximum value parameter of 5.

barrierPenalty(4,-2,2,5)

ans = 5

Calculate the penalty value for the point 0.1 inside the interval [-2,2], using a curvature parameter of 0.5.

barrierPenalty(0.1,-2,2,5,0.5)

ans = 0.0013

Calculate the penalty value for the point [-2,0,4] with respect to the box defined by [0,1], [-1,1], and [-2,2] along the x, y, and z dimensions, respectively, using the default value for maximum value and a curvature parameter of 0.

barrierPenalty([-2 0 4],[0 -1 -2],[1 1 2],1,0)

ans = 3×1

     1
     0
     1

Visualize Penalty Values for an Interval

Create a vector of 1001 equidistant points distributed between -5 and 5.

x = -5:0.01:5;

Calculate penalties for all the points in the vector, using the default value for the maximum value parameter and a value of 0.01 for the curvature parameter.

p = barrierPenalty(x,-2,2,1,0.01);

Plot the points and add grid, labels and title.

plot(x,p)
grid
xlabel("point position");
ylabel("penalty value");
title("Penalty values over an interval");

Input Arguments

x — Point for which the penalty is calculated

scalar | vector | matrix

Point for which the penalty is calculated, specified as a numeric scalar, vector or matrix.

Example: [0.5; 1.6]

xmin — Lower bounds

scalar | vector | matrix

Lower bounds for x, specified as a numeric scalar, vector or matrix. To use the same minimum value for all elements in x, specify xmin as a scalar.

Example: -1

xmax — Upper bounds

scalar | vector | matrix

Upper bounds for x, specified as a numeric scalar, vector or matrix. To use the same maximum value for all elements in x, specify xmax as a scalar.

Example: 2

maxValue — Maximum value parameter of the barrier function

1 (default) | nonnegative scalar

Maximum value parameter of the barrier function, specified as a scalar.

Example: 2

curvature — Curvature parameter of the barrier function

0.1 (default) | nonnegative scalar

Curvature parameter of the barrier function, specified as a scalar.

Example: 0.2

Output Arguments

p — Penalty value

nonnegative vector

Penalty value, returned as a vector of nonnegative elements. Each element pi depends on the position of xi with respect to the interval specified by xmini and xmaxi. The barrier penalty function returns the value

$$p(x)=\min \left(p_{\max },C\left(\log \left(0.25{\left(x_{\max }-x_{\min }\right)}^2\right)-\log \left(\left(x-x_{\min }\right)\left(x_{\max }-x_{\min }\right)\right)\right)\right)$$

when x_min < x < x_max, and maxValue otherwise. Here, C is the argument curvature, and p_max is the argument maxValue. Note that for positive values of C the returned penalty value is always positive. If C is zero, then the returned penalty is zero inside the interval defined by the bounds, and p_max outside this interval. If x is multidimensional, then the calculation is applied independently on each dimension. Penalty functions are typically used to generate negative rewards when constraints are violated, such as in generateRewardFunction.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.