Multi-period Value-at-Risk Constrained Portfolio Optimization via DC Programming
TL;DR
A projected inertial Boosted Difference-of-Convex Functions Algorithm (iBDCA) that combines inertial extrapolation, an objective safeguard, and a boosted line search is proposed that is well defined, generates a monotonically decreasing objective sequence, and has only critical accumulation points for the penalized DC problem.
Abstract
We study a multi-period portfolio optimization problem with finite-scenario Value-at-Risk (VaR) constraints, transaction costs, and diversification regularization. Using a finite-scenario VaR--CVaR identity, we derive a penalized difference-of-convex (DC) formulation over the underlying convex portfolio set. To solve the resulting nonsmooth and nonconvex problem, we propose a projected inertial Boosted Difference-of-Convex Functions Algorithm (iBDCA) that combines inertial extrapolation, an objective safeguard, and a boosted line search. We prove that the method is well defined, generates a monotonically decreasing objective sequence, and has only critical accumulation points for the penalized DC problem. Under a local no-ties condition at an accumulation point, we further establish whole sequence convergence with a local \(R\)-linear rate. Numerical experiments compare iBDCA with DCA and standard BDCA on matched problem instances. Out-of-sample backtests additionally include equal-weight and buy-and-hold benchmarks to illustrate the trade-offs among realized return, risk, transaction costs, and empirical VaR control.