A Projected Steepest-Descent Algorithm for the Nonlinear Constrained Optimization of Football Team Performance

Abstract

Coaching staff must repeatedly decide how to divide a finite pool of training and match-day effort across tactical roles, and evidence from load-monitoring research indicates that the resulting gains in synergy and the resulting build-up of fatigue do not scale in proportion to that effort (Mandorino et al., 2025; Harris et al., 2025). Yet the great majority of published squad-composition and lineup-selection models still rely on binary, integer or purely linear-programming formulations (Zhao et al., 2021; Deb and Das, 2023), which cannot represent such curvature. This paper instead formulates team-performance maximisation as a continuously differentiable nonlinear programme whose objective combines power-law direct contributions, pairwise synergistic interactions between roles, and quadratic fatigue penalties, subject to individual workload ceilings, a global squad-load budget, a nonlinear fatigue-interaction inequality and an effort-sum equality constraint. A projected steepest-descent algorithm equipped with an Armijo backtracking line search and an exact Euclidean projection is developed, analysed and implemented in Python using NumPy, SciPy and automatic differentiation via JAX. On calibrated professional-season data the algorithm converges in 47 iterations to a stationary allocation that improves performance by 9.9% relative to a uniform allocation and by 6.2% relative to the club's historical average. A Karush–Kuhn–Tucker multiplier analysis shows that only two constraints bind at the optimum — the global load budget and the pairwise fatigue-interaction inequality — while all five individual role ceilings remain slack. Scenario and parametric sensitivity experiments confirm the robustness of this qualitative structure and show that a linearised counterpart of the same problem underperforms the nonlinear optimum by 3.6% while misidentifying the operative bottleneck. The resulting Lagrange multipliers translate directly into prioritised, quantitatively grounded recommendations for rotation policy and midfield-freshness preservation, illustrating the practical value of a transparent first-order method as decision support for coaching staff.