Two-Time-Scale Weight Adaptation and Evolutionary Genome Search for Pareto Front Coverage

Many problems in science and engineering require minimising several conflicting objectives at once. In settings where gradients of the objectives are available, e.g. multi-task learning, preference alignment, and adjoint-based design, scalarisation reduces the problem to a gradient-friendly family of weighted sub-problems. However, it shifts the problem onto weight selection: the weight distribution that spreads solutions evenly over the Pareto front is governed by the solution map that is non-injective and often discontinuous. We propose TWARD (Two-time-scale Weight Adaptation and Repulsion Descent), a two-time-scale method for gradient-based Multi-objective optimisation. On the fast scale, the decision variables descend an entropic reference-point scalarisation, a smoothed Tchebycheff achievement function whose reference point is parametrised by the log of the weights, with a parameter $\tau$ which controls coverage. On the slow scale, entropic mirror descent moves the weights on the simplex under a repulsion measured between solutions in objective space, promoting spreading of solutions on the Pareto front, translated into a weight update in closed form by inverting the scalarisation rather than the solution map. Where a single gradient flow is confined to one connected piece of the front, an evolutionary outer loop varies the $m$-dimensional scalarisation genome $(\lambda ,\tau )$. At a matched evaluation budget, we see superior performance compared to established baseline approaches over a wide range of synthetic and real-world tasks.