Convex and sphere packing approaches to portfolio optimization

Classical mean–variance optimization yields exact efficient-frontier allocations for prescribed return or risk levels; however, such extremal solutions may be fragile under estimation error and mandate perturbations. This paper studies portfolio construction within the quadratic mean–variance framework from a feasibility and structural stability perspective by restricting attention to $ \varepsilon $–approximate admissible allocations under return, variance, and relaxed budget constraints. Benchmark consistent feasibility restoration is formulated as a convex quadratic programming problem that computes the allocation closest to a reference portfolio while preserving the quadratic structure and guaranteeing existence and uniqueness of the admissible solution. To quantify interior stability, we introduce a sphere–packing–based geometric construction that inscribes the largest Euclidean ball within the linear return–budget region, yielding the Chebyshev center and an explicit robustness radius equal to the distance to the constraint boundary. The quadratic variance constraint is subsequently incorporated through analytic directional feasibility adjustment, ensuring membership in the intersection of linear and variance admissible sets without extremizing expected return or variance. Unlike ambiguity based or distributionally robust formulations, robustness is defined intrinsically through geometric interior maximization within the prescribed feasible region. Computational experiments calibrated to the investment mandate of the Sovereign Wealth Fund of Mongolia illustrate stable feasibility restoration and robustness under parameter perturbations, establishing sphere–packing–based interior maximization as a complementary geometric robustness principle within the classical mean–variance framework.