Maximum Sharpe Ratio: Declare the Benchmark and Denominator

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export interface MaximumSharpeResult {
    status: string;
    variant: string;
    method: string;
    branch: string | null;
    assetIds: string[] | null;
    mu: number[] | null;
    riskFreeReturn: number | null;
    weights: number[] | null;
    expectedReturn: number | null;
    excessReturn: number | null;
    variance: number | null;
    volatility: number | null;
    sharpe: number | null;
    budgetResidual: number | null;
    lowerBoundResidual: number | null;
    iterations: number;
    maxIterations: number;
    solutionClass: string;
    diagnostics: Record<string, unknown>;
    warnings: string[];
}
const DEFAULT_MAX_ITERATIONS = 10000;
const MAX_SUPPORT_ASSETS = 12;
const TOL_SYMMETRY = 1e-12;
const TOL_PSD = 1e-10;
const TOL_PD = 1e-10;
const TOL_EIGEN_RESIDUAL = 1e-12;
const TOL_OBJECTIVE = 1e-10;
function objectiveClose(left: number, right: number): boolean {
    return Math.abs(left - right) <= TOL_OBJECTIVE * Math.max(Math.abs(left), Math.abs(right), 1e-300);
}
function finiteReal(value: unknown): value is number {
    return typeof value === "number" && Number.isFinite(value);
}
function dot(left: number[], right: number[]): number {
    return left.reduce((sum, value, index) => sum + value * right[index], 0);
}
function matVec(matrix: number[][], vector: number[]): number[] {
    return matrix.map((row) => dot(row, vector));
}
function quadratic(matrix: number[][], vector: number[]): number {
    return dot(vector, matVec(matrix, vector));
}
function maxAbs(matrix: number[][]): number {
    return Math.max(0, ...matrix.flat().map((value) => Math.abs(value)));
}
function jacobiEigenvalues(matrix: number[][]): {
    values: number[];
    residual: number;
} {
    const size = matrix.length;
    if (size === 1)
        return { values: [matrix[0][0]], residual: 0 };
    const work = matrix.map((row) => row.slice());
    const maxSweeps = Math.max(32, 8 * size * size);
    for (let sweep = 0; sweep < maxSweeps; sweep += 1) {
        let p = 0;
        let q = 1;
        let largest = 0;
        for (let i = 0; i < size; i += 1) {
            for (let j = i + 1; j < size; j += 1) {
                if (Math.abs(work[i][j]) > largest) {
                    largest = Math.abs(work[i][j]);
                    p = i;
                    q = j;
                }
            }
        }
        if (largest <= 1e-15)
            break;
        const angle = 0.5 * Math.atan2(2 * work[p][q], work[q][q] - work[p][p]);
        const cosine = Math.cos(angle);
        const sine = Math.sin(angle);
        for (let k = 0; k < size; k += 1) {
            if (k === p || k === q)
                continue;
            const pk = work[p][k];
            const qk = work[q][k];
            work[p][k] = cosine * pk - sine * qk;
            work[k][p] = work[p][k];
            work[q][k] = sine * pk + cosine * qk;
            work[k][q] = work[q][k];
        }
        const pp = work[p][p];
        const qq = work[q][q];
        const pq = work[p][q];
        work[p][p] = cosine * cosine * pp - 2 * sine * cosine * pq + sine * sine * qq;
        work[q][q] = sine * sine * pp + 2 * sine * cosine * pq + cosine * cosine * qq;
        work[p][q] = 0;
        work[q][p] = 0;
    }
    let residual = 0;
    for (let i = 0; i < size; i += 1) {
        for (let j = 0; j < size; j += 1) {
            if (i !== j)
                residual = Math.max(residual, Math.abs(work[i][j]));
        }
    }
    return { values: work.map((row, index) => row[index]), residual };
}
function solveLinear(matrix: number[][], vector: number[]): number[] | null {
    const size = vector.length;
    const augmented = matrix.map((row, index) => [...row, vector[index]]);
    for (let column = 0; column < size; column += 1) {
        let pivot = column;
        for (let row = column + 1; row < size; row += 1) {
            if (Math.abs(augmented[row][column]) > Math.abs(augmented[pivot][column]))
                pivot = row;
        }
        if (Math.abs(augmented[pivot][column]) <= 1e-14)
            return null;
        [augmented[column], augmented[pivot]] = [augmented[pivot], augmented[column]];
        const divisor = augmented[column][column];
        for (let j = column; j <= size; j += 1)
            augmented[column][j] /= divisor;
        for (let row = 0; row < size; row += 1) {
            if (row === column)
                continue;
            const factor = augmented[row][column];
            if (factor === 0)
                continue;
            for (let j = column; j <= size; j += 1)
                augmented[row][j] -= factor * augmented[column][j];
        }
    }
    return augmented.map((row) => row[size]);
}
function combinations(size: number, choose: number): number[][] {
    const output: number[][] = [];
    const current: number[] = [];
    const visit = (start: number): void => {
        if (current.length === choose) {
            output.push(current.slice());
            return;
        }
        for (let index = start; index <= size - (choose - current.length); index += 1) {
            current.push(index);
            visit(index + 1);
            current.pop();
        }
    };
    visit(0);
    return output;
}
function failure(status: string, message: string, details: Partial<MaximumSharpeResult> = {}, diagnostics: Record<string, unknown> = {}): MaximumSharpeResult {
    return {
        status,
        variant: "maximum-sharpe-long-only-pd",
        method: "exhaustive-support-enumeration",
        branch: null,
        assetIds: details.assetIds ?? null,
        mu: details.mu ?? null,
        riskFreeReturn: details.riskFreeReturn ?? null,
        weights: null,
        expectedReturn: null,
        excessReturn: null,
        variance: null,
        volatility: null,
        sharpe: null,
        budgetResidual: null,
        lowerBoundResidual: null,
        iterations: 0,
        maxIterations: details.maxIterations ?? DEFAULT_MAX_ITERATIONS,
        solutionClass: "unknown",
        diagnostics: { message, ...diagnostics },
        warnings: [],
    };
}
function result(assetIds: string[], mu: number[], riskFreeReturn: number, weights: number[], covariance: number[][], covarianceScale: number, branch: string, iterations: number, maxIterations: number, solutionClass: string, diagnostics: Record<string, unknown>): MaximumSharpeResult {
    const normalizedVariance = quadratic(covariance, weights);
    const variance = normalizedVariance * covarianceScale;
    if (!Number.isFinite(variance) || variance <= 0)
        return failure("numerical_issue", "the selected covariance scale produced no finite positive denominator.", { assetIds, mu, riskFreeReturn }, { ...diagnostics, covarianceScale, variance });
    const volatility = Math.sqrt(variance);
    const expectedReturn = dot(mu, weights);
    const excessReturn = mu.reduce((sum, value, index) => sum + (value - riskFreeReturn) * weights[index], 0);
    if (!Number.isFinite(expectedReturn) || !Number.isFinite(excessReturn))
        return failure("numerical_issue", "expected or excess return overflowed the finite arithmetic contract.", { assetIds, mu, riskFreeReturn }, { ...diagnostics, covarianceScale });
    if (!Number.isFinite(excessReturn / volatility))
        return failure("numerical_issue", "the Sharpe ratio is not finite at the selected denominator.", { assetIds, mu, riskFreeReturn }, { ...diagnostics, covarianceScale });
    return {
        status: "optimal",
        variant: "maximum-sharpe-long-only-pd",
        method: "exhaustive-support-enumeration",
        branch,
        assetIds,
        mu,
        riskFreeReturn,
        weights,
        expectedReturn,
        excessReturn,
        variance,
        volatility,
        sharpe: excessReturn / volatility,
        budgetResidual: Math.abs(weights.reduce((sum, value) => sum + value, 0) - 1),
        lowerBoundResidual: Math.max(0, -Math.min(...weights)),
        iterations,
        maxIterations,
        solutionClass,
        diagnostics: { covarianceScale, normalizedVariance, objectiveGap: 0, ...diagnostics },
        warnings: [],
    };
}
export function maximumSharpe(assetIdsInput: unknown, muInput: unknown, covarianceInput: unknown, riskFreeReturnInput: unknown, options: {
    maxIterations?: unknown;
} = {}): MaximumSharpeResult {
    if (!Array.isArray(assetIdsInput))
        return failure("invalid_input", "assetIds must be an ordered array.");
    const assetIds = assetIdsInput.slice();
    if (!assetIds.length || assetIds.some((value) => typeof value !== "string" || value.trim() === ""))
        return failure("invalid_input", "assetIds must contain non-empty strings.");
    if (new Set(assetIds).size !== assetIds.length)
        return failure("invalid_input", "assetIds must be unique.");
    if (!Array.isArray(muInput) || muInput.length !== assetIds.length || muInput.some((value) => !finiteReal(value)))
        return failure("invalid_input", "mu must be a finite vector aligned to assetIds.", { assetIds });
    const mu = muInput.slice() as number[];
    if (!finiteReal(riskFreeReturnInput))
        return failure("invalid_input", "riskFreeReturn must be a finite same-horizon return.", { assetIds, mu });
    const riskFreeReturn = riskFreeReturnInput;
    const maxIterations = options.maxIterations ?? DEFAULT_MAX_ITERATIONS;
    if (!Number.isInteger(maxIterations) || (maxIterations as number) < 0 || (maxIterations as number) > DEFAULT_MAX_ITERATIONS)
        return failure("invalid_input", "maxIterations must be an integer from 0 through 10,000.", { assetIds, mu, riskFreeReturn });
    const iterationBudget = maxIterations as number;
    if (!Array.isArray(covarianceInput) || covarianceInput.length !== assetIds.length || covarianceInput.some((row) => !Array.isArray(row) || row.length !== assetIds.length))
        return failure("invalid_input", "covariance must be a square matrix aligned to assetIds.", { assetIds, mu, riskFreeReturn });
    if (covarianceInput.some((row) => (row as unknown[]).some((value) => !finiteReal(value))))
        return failure("invalid_input", "covariance entries must be finite real numbers.", { assetIds, mu, riskFreeReturn });
    const covariance = (covarianceInput as unknown[][]).map((row) => row.slice() as number[]);
    const scale = maxAbs(covariance);
    if (scale === 0)
        return failure("unsupported_singular_covariance", "zero covariance has a zero-risk denominator.", { assetIds, mu, riskFreeReturn }, { covarianceScale: 0 });
    const normalized = covariance.map((row) => row.map((value) => value / scale));
    let symmetryResidual = 0;
    for (let i = 0; i < assetIds.length; i += 1)
        for (let j = 0; j < assetIds.length; j += 1)
            symmetryResidual = Math.max(symmetryResidual, Math.abs(normalized[i][j] - normalized[j][i]));
    if (symmetryResidual > TOL_SYMMETRY)
        return failure("invalid_input", "covariance symmetry residual exceeds the normalized tolerance.", { assetIds, mu, riskFreeReturn }, { covarianceScale: scale, symmetryResidualS: symmetryResidual });
    const symmetric = normalized.map((row, i) => row.map((value, j) => 0.5 * (value + normalized[j][i])));
    const eigen = jacobiEigenvalues(symmetric);
    const minimumEigenvalue = Math.min(...eigen.values);
    const baseDiagnostics = { covarianceScale: scale, symmetryResidualS: symmetryResidual, minimumEigenvalueS: minimumEigenvalue, eigenResidualS: eigen.residual, tolPsd: TOL_PSD, tolPd: TOL_PD };
    if (eigen.residual > TOL_EIGEN_RESIDUAL)
        return failure("numerical_issue", "eigensolver residual exceeds the normalized tolerance.", { assetIds, mu, riskFreeReturn }, baseDiagnostics);
    if (minimumEigenvalue < -TOL_PSD)
        return failure("invalid_input", "covariance is materially indefinite.", { assetIds, mu, riskFreeReturn }, baseDiagnostics);
    if (minimumEigenvalue <= TOL_PD)
        return failure("unsupported_singular_covariance", "this ratio core requires positive-definite covariance.", { assetIds, mu, riskFreeReturn }, baseDiagnostics);
    if (iterationBudget === 0)
        return failure("numerical_issue", "maxIterations=0 cannot produce a ratio certificate.", { assetIds, mu, riskFreeReturn, maxIterations: iterationBudget }, baseDiagnostics);
    const size = assetIds.length;
    if (size > MAX_SUPPORT_ASSETS)
        return failure("numerical_issue", "support enumeration is bounded at 12 assets in this deterministic core.", { assetIds, mu, riskFreeReturn }, { ...baseDiagnostics, maxSupportAssets: MAX_SUPPORT_ASSETS });
    const excess = mu.map((value) => value - riskFreeReturn);
    const diagonalVolatility = symmetric.map((row, index) => Math.sqrt(row[index]));
    if (excess.every((value) => value <= 0)) {
        const ratios = excess.map((value, index) => value / diagonalVolatility[index]);
        const bestRatio = Math.max(...ratios);
        const bestIndex = ratios.indexOf(bestRatio);
        const ties = ratios.filter((value) => objectiveClose(value, bestRatio));
        return result(assetIds, mu, riskFreeReturn, ratios.map((_value, index) => index === bestIndex ? 1 : 0), symmetric, scale, "nonpositive_excess_endpoint", 1, iterationBudget, ties.length > 1 ? "non_unique" : "unique", { ...baseDiagnostics, endpointRatios: ratios, certificate: "endpoint-excess-over-volatility-bound" });
    }
    let bestRatio = -Infinity;
    let bestWeights: number[] | null = null;
    let bestSupport: number[] = [];
    let tiedSupports = 0;
    let supportCount = 0;
    const supportsTotal = (2 ** size) - 1;
    let checkedSupports = 0;
    for (let supportSize = 1; supportSize <= size; supportSize += 1) {
        for (const support of combinations(size, supportSize)) {
            if (checkedSupports >= iterationBudget)
                break;
            checkedSupports += 1;
            const supportMatrix = support.map((i) => support.map((j) => symmetric[i][j]));
            const direction = solveLinear(supportMatrix, support.map((i) => excess[i]));
            if (direction === null || direction.some((value) => value <= 0))
                continue;
            const total = direction.reduce((sum, value) => sum + value, 0);
            if (total <= 0)
                continue;
            const weights = Array.from({ length: size }, (_value, index) => { const position = support.indexOf(index); return position < 0 ? 0 : direction[position] / total; });
            const variance = quadratic(symmetric, weights);
            const numerator = dot(excess, weights);
            if (variance <= 0 || numerator <= 0)
                continue;
            const ratio = numerator / Math.sqrt(variance);
            supportCount += 1;
            if (bestWeights === null || (ratio > bestRatio && !objectiveClose(ratio, bestRatio))) {
                bestRatio = ratio;
                bestWeights = weights;
                bestSupport = support;
                tiedSupports = 1;
            }
            else if (objectiveClose(ratio, bestRatio))
                tiedSupports += 1;
        }
        if (checkedSupports >= iterationBudget)
            break;
    }
    if (checkedSupports < supportsTotal)
        return failure("numerical_issue", "maxIterations exhausted before all supports were checked.", { assetIds, mu, riskFreeReturn, iterations: checkedSupports, maxIterations: iterationBudget }, { ...baseDiagnostics, supportsChecked: checkedSupports, supportsRequired: supportsTotal, positiveExcessAssets: excess.filter((value) => value > 0).length });
    if (bestWeights === null)
        return failure("numerical_issue", "no positive-excess support produced a certified ratio.", { assetIds, mu, riskFreeReturn }, { ...baseDiagnostics, certificate: "not-certified", positiveExcessAssets: excess.filter((value) => value > 0).length });
    return result(assetIds, mu, riskFreeReturn, bestWeights, symmetric, scale, "positive_excess_support", checkedSupports, iterationBudget, tiedSupports > 1 ? "non_unique" : "unique", { ...baseDiagnostics, certificate: "exhaustive-support-enumeration", support: bestSupport.map((index) => assetIds[index]), supportsChecked: checkedSupports, supportsRequired: supportsTotal, supportsCertified: supportCount, objectiveGap: 0 });
}
export const maximumSharpeRatio = maximumSharpe;