Numerical methods for lambda quantiles: robust evaluation and portfolio optimisation

Lambda quantiles, originally introduced as lambda value at risk, generalise the classical value at risk by allowing for a variable confidence level. This work presents efficient algorithms for computing lambda quantiles and demonstrates their application in portfolio optimisation. We first develop a robust algorithm, {\Lambda}-Newton-Bis, that combines Newton's method with a bisection strategy to ensure global convergence. The algorithm handles potential discontinuities and achieves local quadratic convergence under standard regularity assumptions. To address cases with multiple roots, we also propose an interval analysis approach. We then demonstrate the algorithm's computational efficiency and practical relevance within a portfolio optimization framework. To this end, we develop two alternative solution methods that incorporate the {\Lambda}-Newton-Bis procedure. Numerical experiments confirm the algorithm's convergence properties and highlight its computational advantages in optimization tasks based on lambda quantiles.