Turnover-Constrained Optimization: A Hard Rebalance Budget From Current Holdings

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/**
 * General-N turnover-constrained long-only Markowitz reference solver.
 *
 * This isolated staging implementation solves
 *   maximize mu·w - lambdaRisk * wᵀ Sigma w
 *   subject to w >= 0, sum(w) = 1, 0.5*||w-w0||1 <= tau.
 *
 * The projection is onto the simplex intersected with the L1 ball; it does
 * not clip weights and silently renormalize them.  The returned certificate
 * is a projected-KKT residual.  A max-iteration result is never optimal.
 */

export const COMPARISON_ABSOLUTE_TOLERANCE = 1e-9;
export const COMPARISON_RELATIVE_TOLERANCE = 1e-8;
export const FEASIBILITY_TOLERANCE = 1e-10;
export const CURRENT_SUM_TOLERANCE = 1e-12;
export const PSD_RELATIVE_TOLERANCE = 1e-10;
export const STATIONARITY_TOLERANCE = 1e-8;
export const PROJECTION_BISECTION_ITERATIONS = 60;
export const DEFAULT_MAX_ITERATIONS = 5000;

export class TurnoverValidationError extends Error {
  readonly code: string;

  constructor(code: string, message: string) {
    super(message);
    this.name = "TurnoverValidationError";
    this.code = code;
  }
}

export class TurnoverNumericalError extends Error {
  readonly code: string;

  constructor(code: string, message: string) {
    super(message);
    this.name = "TurnoverNumericalError";
    this.code = code;
  }
}

export interface TurnoverCertificate {
  kind: "projected-gradient-kkt-plus-frank-wolfe-gap";
  stationarityResidual: number;
  frankWolfeGap: number;
  objectiveGapUpperBound: number;
  normalizedStationarityResidual: number;
  normalizedFrankWolfeGap: number;
  normalizedObjectiveGapUpperBound: number;
  objectiveScale: number;
  stationarityTolerance: number;
  toleranceUnits: string;
  lipschitzUpperBound: number;
  stepDenominator: number;
  stepSize: number;
  turnoverRadius: number;
  globalOptimality: string;
}

export interface TurnoverResult {
  status: "optimal" | "max-iterations";
  weights: number[];
  objective: number;
  directTurnover: number;
  inputCurrentSum: number;
  simplexResidual: number;
  minWeight: number;
  iterations: number;
  frankWolfeGap: number;
  certificate: TurnoverCertificate;
}

function invalid(code: string, message: string): never {
  throw new TurnoverValidationError(code, message);
}

function asReal(value: unknown, name: string): number {
  if (typeof value !== "number" || !Number.isFinite(value)) {
    invalid("invalid-input", `${name} must be a finite real number.`);
  }
  return value;
}

function vector(values: unknown, name: string): number[] {
  if (!Array.isArray(values) || values.length === 0) {
    invalid("invalid-input", `${name} must be a non-empty numeric array.`);
  }
  return values.map((value, index) => asReal(value, `${name}[${index}]`));
}

function matrix(values: unknown, name: string, size: number): number[][] {
  if (!Array.isArray(values) || values.length !== size) {
    invalid("invalid-input", `${name} must have ${size} rows.`);
  }
  return values.map((row, rowIndex) => {
    if (!Array.isArray(row) || row.length !== size) {
      invalid("invalid-input", `${name} must be ${size} by ${size}.`);
    }
    return row.map((value, columnIndex) => asReal(value, `${name}[${rowIndex}][${columnIndex}]`));
  });
}

function stableSum(values: readonly number[]): number {
  let total = 0;
  let compensation = 0;
  for (const value of values) {
    const candidate = total + value;
    if (Math.abs(total) >= Math.abs(value)) {
      compensation += (total - candidate) + value;
    } else {
      compensation += (value - candidate) + total;
    }
    total = candidate;
  }
  return total + compensation;
}

function dot(left: readonly number[], right: readonly number[]): number {
  return stableSum(left.map((value, index) => value * right[index]));
}

function matVec(matrixValue: readonly (readonly number[])[], vectorValue: readonly number[]): number[] {
  return matrixValue.map((row) => dot(row, vectorValue));
}

function quadratic(matrixValue: readonly (readonly number[])[], vectorValue: readonly number[]): number {
  return dot(vectorValue, matVec(matrixValue, vectorValue));
}

function l1Distance(left: readonly number[], right: readonly number[]): number {
  return stableSum(left.map((value, index) => Math.abs(value - right[index])));
}

function minimumSymmetricEigenvalue(matrixValue: readonly (readonly number[])[]): number {
  const size = matrixValue.length;
  const work = matrixValue.map((row) => [...row]);
  if (size === 1) return work[0][0];
  const scale = Math.max(1e-300, ...work.flat().map((value) => Math.abs(value)));
  const maxSweeps = Math.max(50, 20 * size * size);
  for (let sweep = 0; sweep < maxSweeps; sweep += 1) {
    let p = 0;
    let q = 1;
    let largest = 0;
    for (let row = 0; row < size; row += 1) {
      for (let column = row + 1; column < size; column += 1) {
        const magnitude = Math.abs(work[row][column]);
        if (magnitude > largest) {
          largest = magnitude;
          p = row;
          q = column;
        }
      }
    }
    if (largest <= 1e-14 * scale) break;
    const app = work[p][p];
    const aqq = work[q][q];
    const apq = work[p][q];
    const angle = 0.5 * Math.atan2(2 * apq, aqq - app);
    const cosine = Math.cos(angle);
    const sine = Math.sin(angle);
    for (let index = 0; index < size; index += 1) {
      if (index === p || index === q) continue;
      const aip = work[index][p];
      const aiq = work[index][q];
      work[index][p] = cosine * aip - sine * aiq;
      work[p][index] = work[index][p];
      work[index][q] = sine * aip + cosine * aiq;
      work[q][index] = work[index][q];
    }
    work[p][p] = cosine * cosine * app - 2 * cosine * sine * apq + sine * sine * aqq;
    work[q][q] = sine * sine * app + 2 * cosine * sine * apq + cosine * cosine * aqq;
    work[p][q] = 0;
    work[q][p] = 0;
  }
  return Math.min(...work.map((row, index) => row[index]));
}

interface ValidatedInputs {
  mean: number[];
  sigma: number[][];
  riskAversion: number;
  current: number[];
  turnoverLimit: number;
}

function validateInputs(
  mu: unknown,
  covariance: unknown,
  lambdaRisk: unknown,
  w0: unknown,
  tau: unknown,
): ValidatedInputs {
  const mean = vector(mu, "mu");
  const size = mean.length;
  const sigma = matrix(covariance, "covariance", size);
  const riskAversion = asReal(lambdaRisk, "lambdaRisk");
  const current = vector(w0, "w0");
  const turnoverLimit = asReal(tau, "tau");
  if (current.length !== size) {
    invalid("invalid-input", "mu, covariance, and w0 must have the same dimension.");
  }
  if (riskAversion < 0) invalid("invalid-input", "lambdaRisk must be nonnegative.");
  if (turnoverLimit < 0) invalid("invalid-input", "tau must be nonnegative.");
  for (let row = 0; row < size; row += 1) {
    for (let column = 0; column < row; column += 1) {
      if (Math.abs(sigma[row][column] - sigma[column][row]) > 1e-12) {
        invalid("indefinite-covariance", "covariance must be symmetric within tolerance.");
      }
    }
  }
  const matrixScale = Math.max(1e-300, ...sigma.flat().map((value) => Math.abs(value)));
  if (minimumSymmetricEigenvalue(sigma) < -PSD_RELATIVE_TOLERANCE * matrixScale) {
    invalid("indefinite-covariance", "covariance must be positive semidefinite.");
  }
  if (current.some((weight) => weight < -FEASIBILITY_TOLERANCE)) {
    invalid("invalid-input", "w0 must be nonnegative.");
  }
  const currentSum = stableSum(current);
  if (Math.abs(currentSum - 1) > CURRENT_SUM_TOLERANCE) {
    invalid("invalid-input", "w0 must sum to one within the declared tolerance.");
  }
  // Do not silently renormalize the current portfolio; retain the validated
  // input so any tiny residue remains visible to the caller.
  return {mean, sigma, riskAversion, current, turnoverLimit};
}

function simplexProjection(values: readonly number[]): number[] {
  const indexed = values.map((value, index) => ({value, index}));
  indexed.sort((left, right) => right.value - left.value);
  let cumulative = 0;
  let rho = -1;
  for (let index = 0; index < indexed.length; index += 1) {
    cumulative += indexed[index].value;
    const candidate = indexed[index].value - (cumulative - 1) / (index + 1);
    if (candidate > 0) rho = index;
  }
  if (rho < 0) rho = 0;
  const threshold = (stableSum(indexed.slice(0, rho + 1).map((item) => item.value)) - 1) / (rho + 1);
  return values.map((value) => Math.max(value - threshold, 0));
}

function l1SimplexProx(values: readonly number[], center: readonly number[], penalty: number): number[] {
  const scale = Math.max(1, penalty, ...values.map((value) => Math.abs(value)), ...center.map((value) => Math.abs(value)));
  const valuesAt = (nu: number): number[] => values.map((value, index) => {
    const anchor = center[index];
    const shifted = value - nu;
    if (shifted > anchor + penalty) return shifted - penalty;
    if (shifted >= anchor - penalty) return anchor;
    return Math.max(0, shifted + penalty);
  });
  let lower = Math.min(...values) - scale - penalty - 1;
  let upper = Math.max(...values) + scale + penalty + 1;
  for (let index = 0; index < 80 && stableSum(valuesAt(lower)) < 1; index += 1) lower -= scale;
  for (let index = 0; index < 80 && stableSum(valuesAt(upper)) > 1; index += 1) upper += scale;
  for (let index = 0; index < PROJECTION_BISECTION_ITERATIONS; index += 1) {
    const middle = (lower + upper) / 2;
    if (stableSum(valuesAt(middle)) > 1) lower = middle;
    else upper = middle;
  }
  return valuesAt((lower + upper) / 2);
}

function projectTurnoverSimplex(values: readonly number[], center: readonly number[], radius: number): number[] {
  const simplex = simplexProjection(values);
  if (l1Distance(simplex, center) <= radius + FEASIBILITY_TOLERANCE) return simplex;
  if (radius <= FEASIBILITY_TOLERANCE) return [...center];
  let lowerPenalty = 0;
  let upperPenalty = 1;
  let candidate = l1SimplexProx(values, center, upperPenalty);
  for (let index = 0; index < 80 && l1Distance(candidate, center) > radius; index += 1) {
    upperPenalty *= 2;
    candidate = l1SimplexProx(values, center, upperPenalty);
  }
  for (let index = 0; index < PROJECTION_BISECTION_ITERATIONS; index += 1) {
    const middle = (lowerPenalty + upperPenalty) / 2;
    candidate = l1SimplexProx(values, center, middle);
    if (l1Distance(candidate, center) > radius) lowerPenalty = middle;
    else upperPenalty = middle;
  }
  return l1SimplexProx(values, center, upperPenalty);
}

function objective(mean: readonly number[], sigma: readonly (readonly number[])[], riskAversion: number, weights: readonly number[]): number {
  return dot(mean, weights) - riskAversion * quadratic(sigma, weights);
}

function turnoverLinearOracle(
  gradient: readonly number[],
  current: readonly number[],
  turnoverLimit: number,
): number[] {
  /** Maximize a linear objective over the fixed simplex/L1 feasible set. */
  const result = [...current];
  if (turnoverLimit <= FEASIBILITY_TOLERANCE) return result;
  const donorRemaining = [...current];
  const receiverRemaining = current.map((value) => 1 - value);
  const donorOrder = current.map((_, index) => index)
    .sort((left, right) => gradient[left] - gradient[right] || left - right);
  const receiverOrder = current.map((_, index) => index)
    .sort((left, right) => gradient[right] - gradient[left] || left - right);
  let remainingTurnover = Math.min(turnoverLimit, 1);
  for (const receiver of receiverOrder) {
    if (remainingTurnover <= FEASIBILITY_TOLERANCE) break;
    if (receiverRemaining[receiver] <= FEASIBILITY_TOLERANCE) continue;
    for (const donor of donorOrder) {
      if (remainingTurnover <= FEASIBILITY_TOLERANCE) break;
      if (donor === receiver || donorRemaining[donor] <= FEASIBILITY_TOLERANCE) continue;
      const gain = gradient[receiver] - gradient[donor];
      if (gain <= 0) break;
      const amount = Math.min(remainingTurnover, donorRemaining[donor], receiverRemaining[receiver]);
      if (amount <= FEASIBILITY_TOLERANCE) continue;
      result[donor] -= amount;
      result[receiver] += amount;
      donorRemaining[donor] -= amount;
      receiverRemaining[receiver] -= amount;
      remainingTurnover -= amount;
    }
  }
  return result;
}

export interface TurnoverSolverOptions {
  maxIterations?: number;
  stationarityTolerance?: number;
}

export function solveTurnoverConstrainedMarkowitz(
  mu: unknown,
  covariance: unknown,
  lambdaRisk: unknown,
  w0: unknown,
  tau: unknown,
  options: TurnoverSolverOptions = {},
): TurnoverResult {
  const {mean, sigma, riskAversion, current, turnoverLimit} = validateInputs(
    mu, covariance, lambdaRisk, w0, tau,
  );
  const maxIterations = options.maxIterations ?? DEFAULT_MAX_ITERATIONS;
  const stationarityTolerance = options.stationarityTolerance ?? STATIONARITY_TOLERANCE;
  if (!Number.isInteger(maxIterations) || maxIterations <= 0) {
    invalid("invalid-input", "maxIterations must be a positive integer.");
  }
  if (!Number.isFinite(stationarityTolerance) || stationarityTolerance <= 0) {
    invalid("invalid-input", "stationarityTolerance must be positive and finite.");
  }

  const size = mean.length;
  const hScale = 2 * riskAversion;
  const hLipschitz = Math.max(
    0,
    ...sigma.map((row) => stableSum(row.map((value) => Math.abs(hScale * value)))),
  );
  const objectiveScale = Math.max(
    1e-300,
    ...mean.map((value) => Math.abs(value)),
    riskAversion * Math.max(...sigma.flat().map((value) => Math.abs(value))),
  );
  // A scale-aware step keeps a tiny-but-nonzero linear objective from being
  // mistaken for a zero objective by an absolute unit step/tolerance.
  const stepDenominator = Math.max(hLipschitz, objectiveScale);
  const step = 1 / stepDenominator;
  const radius = 2 * turnoverLimit;
  let weights = [...current];
  let previousObjective = objective(mean, sigma, riskAversion, weights);
  let status: TurnoverResult["status"] = "max-iterations";
  let iterations = 0;
  let stationarity = Number.POSITIVE_INFINITY;
  let frankWolfeGap = Number.POSITIVE_INFINITY;

  for (let iteration = 1; iteration <= maxIterations; iteration += 1) {
    const sigmaWeights = matVec(sigma, weights);
    const gradient = mean.map((value, index) => value - hScale * sigmaWeights[index]);
    const candidate = projectTurnoverSimplex(
      weights.map((weight, index) => weight + step * gradient[index]),
      current,
      radius,
    );
    stationarity = Math.max(...weights.map((weight, index) => Math.abs(weight - candidate[index]))) / step;
    // The linear oracle ranges over the fixed feasible set centered at the
    // original portfolio, not a fresh turnover ball around the iterate.
    const linearOracle = turnoverLinearOracle(gradient, current, turnoverLimit);
    frankWolfeGap = Math.max(
      0,
      dot(gradient, linearOracle.map((value, index) => value - weights[index])),
    );
    const candidateObjective = objective(mean, sigma, riskAversion, candidate);
    if (!Number.isFinite(candidateObjective) || candidate.some((value) => !Number.isFinite(value))) {
      throw new TurnoverNumericalError("numerical-failure", "solver produced a non-finite iterate.");
    }
    iterations = iteration;
    weights = candidate;
    const objectiveChange = Math.abs(candidateObjective - previousObjective);
    previousObjective = candidateObjective;
    const normalizedStationarity = stationarity / objectiveScale;
    const normalizedGap = frankWolfeGap / objectiveScale;
    const normalizedObjectiveChange = objectiveChange / objectiveScale;
    if (
      normalizedStationarity <= stationarityTolerance
      && normalizedGap <= stationarityTolerance
      && normalizedObjectiveChange <= stationarityTolerance * 0.1
    ) {
      status = "optimal";
      break;
    }
  }

  // Recompute the certificate at the returned iterate. The loop's gap is
  // evaluated before its update; a stale value must not describe the result.
  const finalSigmaWeights = matVec(sigma, weights);
  const finalGradient = mean.map((value, index) => value - hScale * finalSigmaWeights[index]);
  const finalCandidate = projectTurnoverSimplex(
    weights.map((weight, index) => weight + step * finalGradient[index]),
    current,
    radius,
  );
  stationarity = Math.max(...weights.map((weight, index) => Math.abs(weight - finalCandidate[index]))) / step;
  const finalOracle = turnoverLinearOracle(finalGradient, current, turnoverLimit);
  frankWolfeGap = Math.max(
    0,
    dot(finalGradient, finalOracle.map((value, index) => value - weights[index])),
  );

  const finalObjective = objective(mean, sigma, riskAversion, weights);
  const directTurnover = 0.5 * l1Distance(weights, current);
  const inputCurrentSum = stableSum(current);
  const simplexResidual = Math.abs(stableSum(weights) - 1);
  const minWeight = Math.min(...weights);
  if (![...weights, finalObjective, directTurnover].every(Number.isFinite)) {
    throw new TurnoverNumericalError("numerical-failure", "final solver diagnostics are non-finite.");
  }
  if (simplexResidual > FEASIBILITY_TOLERANCE || minWeight < -FEASIBILITY_TOLERANCE) {
    throw new TurnoverNumericalError("feasibility-residual", "final weights violate simplex feasibility.");
  }
  if (directTurnover > turnoverLimit + FEASIBILITY_TOLERANCE) {
    throw new TurnoverNumericalError("feasibility-residual", "final weights violate the direct turnover cap.");
  }

  return {
    status,
    weights,
    objective: finalObjective,
    directTurnover,
    inputCurrentSum,
    simplexResidual,
    minWeight,
    iterations,
    frankWolfeGap,
    certificate: {
      kind: "projected-gradient-kkt-plus-frank-wolfe-gap",
      stationarityResidual: stationarity,
      frankWolfeGap,
      objectiveGapUpperBound: frankWolfeGap,
      normalizedStationarityResidual: stationarity / objectiveScale,
      normalizedFrankWolfeGap: frankWolfeGap / objectiveScale,
      normalizedObjectiveGapUpperBound: frankWolfeGap / objectiveScale,
      objectiveScale,
      stationarityTolerance,
      toleranceUnits: "normalized by objectiveScale; original-unit gap is retained above",
      lipschitzUpperBound: hLipschitz,
      stepDenominator,
      stepSize: step,
      turnoverRadius: radius,
      globalOptimality: "For PSD covariance, the Frank-Wolfe linearization gap upper-bounds the concave-QP objective gap; status optimal requires both that gap and the projected-KKT residual within tolerance. max-iterations is not a certificate.",
    },
  };
}