Argmin Inference

Name

UxHwDoubleArgmin, UxHwFloatArgmin — Perform $\mathrm{argmin}$ inference with distributional arithmetic.

Synopsis

#include <uxhw.h>

float UxHwFloatArgmin(
  float     functionOutput,
  float *   argumentArray,
  size_t    numberOfArguments,
  float *   minimizingArgumentInstanceArray
);

double UxHwDoubleArgmin(
  double    functionOutput,
  double *  argumentArray,
  size_t    numberOfArguments,
  double *  minimizingArgumentInstanceArray
);

Description

The UxHwDoubleArgmin() function performs $\mathrm{argmin}$ inference. It calculates the instances of the distributions in array argumentArray that produce the minimum value of the input distribution functionOutput. It stores the minimizing instances in the output array minimizingArgumentInstanceArray of length numberOfArguments, and returns the minimum value of the input distribution functionOutput.

Given a function $f : S \rightarrow \mathbb{R}$, the $\mathrm{argmin}$ of $f$ is defined as $\mathrm{argmin}_S f(x) := \{x \in \Omega : f(x) \leq f(s) \forall s \in S \}$, that is, the $\mathrm{argmin}$ of $f$ is the collection of inputs in $S$ that result in the minimum value of $f$.

Parameters

• functionOutput — An input distribution which is a computational result of the distributions in the input array argumentArray. This is the distribution which UxHwDoubleArgmin() and UxHwFloatArgmin() will attempt to minimize.

• argumentArray — An array of input distributions that were used in computing functionOutput. UxHwDoubleArgmin() and UxHwFloatArgmin() will attempt to infer the instances of the distributions in argumentArray that minimize functionOutput..

• numberOfArguments — The size of the input array argumentArray and the output array minimizingArgumentInstanceArray.

• minimizingArgumentInstanceArray — The instances of the distributions in the array argumentArray that minimize the value of functionOutput.

Return values

The minimum value of functionOutput that is obtained when the distributions in the array argumentArray take on their minimizing values (i.e., values returned in minimizingArgumentInstanceArray).

Example 1: Finding the minimal circumference of a rectangle with a fixed area

#include <stddef.h>
#include <stdio.h>
#include <uxhw.h>

int
main(void)
{
  double    area = 25.0;
  double    baseLength = UxHwDoubleUniformDist(0.0, 5.0);
  double    sideLength = area / baseLength;
  double    circumference = 2 * (baseLength + sideLength);
  double    circumferenceMinimizingBaseLength;
  double    minimalCircumference;

  /*
   *    Find the instance `circumferenceMinimizingBaseLength`
   *    of `baseLength` that minimizes `circumference` and 
   *    the corresponding minimal circumference.
   */
  minimalCircumference = UxHwDoubleArgmin(
    circumference,
    &baseLength,
    1,
    &circumferenceMinimizingBaseLength
  );

  printf("Circumference minimizing base length: %lf\n", 
    circumferenceMinimizingBaseLength);
  printf("Minimal circumference: %lf\n", minimalCircumference);

  return 0;
}

Example 2: Finding the maximum area enclosed by a fixed-length boundary

#include <stddef.h>
#include <stdio.h>
#include <uxhw.h>

int
main(void)
{
  double    totalBoundaryLength = 20.0;
  double    baseLength = UxHwDoubleUniformDist(0.0, 10.0);
  double    sideLength = totalBoundaryLength / 2.0 - baseLength;
  double    area = baseLength * sideLength;
  double    areaMaximizingBaseLength;
  double    maximalArea;

  /*
   *    We want to find the maximizer of `area` (instead of 
   *    minimizer), so we pass the negative value of `area`.
   */
  maximalArea = -UxHwDoubleArgmin(
    -area,
    &baseLength,
    1,
    &areaMaximizingBaseLength
  );

  printf("Area maximizing base length: %lf\n", areaMaximizingBaseLength);
  printf("Maximal area: %lf\n", maximalArea);

  return 0;
}