Complexity Bounds for Smooth Convex Multiobjective Optimization
This work addresses theoretical complexity gaps in multiobjective optimization, providing foundational insights for algorithm design, though it is incremental in refining existing bounds.
The paper tackles the problem of finding Pareto stationary points in smooth multiobjective optimization, establishing complexity bounds that include linear convergence for strongly convex objectives and lower bounds of order 1/T and 1/T^2 for convex problems under different method classes.
We study the oracle complexity of finding $\varepsilon$-Pareto stationary points in smooth multiobjective optimization with $m$ objectives. The progress metric is the Pareto stationarity gap $\mathcal{G}(x)$ (the norm of an optimal convex combination of gradients). Our contributions are fourfold. (i) For strongly convex objectives, any span first-order method (iterates lie in the span of past gradients) exhibits linear convergence no faster than $\exp(-Θ(T/\sqrtκ))$ after $T$ oracle calls, where $κ$ is the condition number, implying $Θ(\sqrtκ\log(1/\varepsilon))$ iterations; this matches classical accelerated upper bounds. (ii) For convex problems and oblivious one-step methods (a fixed scalarization with pre-scheduled step sizes), we prove a lower bound of order $1/T$ on the best gradient norm among the first $T$ iterates. (iii) Although accelerated gradient descent is outside this restricted class, it is an oblivious span method and attains the same $1/T$ upper rate on a fixed scalarization. (iv) For convex problems and general span methods with adaptive scalarizations, we establish a universal lower bound of order $1/T^{2}$ on the gradient norm of the final iterate after $T$ steps, highlighting a gap between known upper bounds and worst-case guarantees. All bounds hold on non-degenerate instances with distinct objectives and non-singleton Pareto fronts; rates are stated up to universal constants and natural problem scaling.