Multivariate (ℝ^2) relative minimum brute-force finder

findMinimum.js

// The function we want to find the minimums of
// https://i.imgur.com/GG6uXvD.png
const getValue = (m, n) => m * m + n * n;

// The sampling interval (in ℝ^2)
let mStart = 0;
let nStart = 0;
let mEnd = 1;
let nEnd = 1;

// On every iteration we decrease the sampling interval by this factor
const reductionFactor = 100;

// We compute `4 * reductionFactor` points (per dimension) inside the sampling interval
const samplingPoints = reductionFactor * 4;

// The offset between these points
let delta = (mEnd - mStart) / samplingPoints;

const requiredPrecision = 2e-15;

function findMinimum() {
  let bestValue = Infinity;
  let bestM, bestN;

  // Find best parameters
  for (let m = mStart; m < mEnd; m += delta) {
    for (let n = nStart; n < nEnd; n += delta) {
      const value = getValue(m, n);

      if (value < bestValue) {
        bestValue = value;
        bestM = m;
        bestN = n;
      }
    }
  }

  // Zoom around the found parameters
  const newSize = (mEnd - mStart) / reductionFactor;

  mStart = bestM - newSize / 2;
  mEnd = mStart + newSize;

  nStart = bestN - newSize / 2;
  nEnd = nStart + newSize;

  delta = newSize / samplingPoints;
}

function start() {
  let i = 0;

  while (delta > requiredPrecision) {
    findMinimum();

    console.log(
      `Iteration #${i++} -> m ∈ [${mStart}, ${mEnd}], n ∈ [${nStart}, ${nEnd}]`
    );
  }
}

start();