We consider composite optimization problems featuring nonsmooth, weakly convex objectives composed with linear operators. A novel variable smoothing accelerated gradient algorithm is developed, synthesizing smoothing techniques with first-order methods. The method achieves 𝒪(ε ^-3) iteration complexity for ε -stationary points under standard assumptions. Sparse signal recovery experiments confirm our algorithm outperforms the methods in [5] in terms of both CPU times and number of iterations.
This paper studies the integral representation problem for functionals on function spaces, focusing on the Shilkret integrals. Unlike the linear Lebesgue integral, the Shilkret integral exhibits distinct nonlinear properties. While the representation theory for the Choquet integral has been extensively developed, results for the Shilkret integral remain limited. To address this gap, we establish a Greco-type representation theorem for the Shilkret integral and analyze conditions ensuring the existence of representing measures with desirable continuity and regularity properties. Applications to several concrete function spaces are also discussed.
We define Cesàro spaces on general measure spaces on which there is a pre-order provided. We find conditions under which these are quasi-Banach function spaces. Classical Cesàro and Copson function and sequence spaces as well as some Morrey spaces appear as special cases of our general construction. We show that, under natural assumptions, the spaces are strictly convex, posses the diameter two property and contain asymptotically isometric copies of ℓ _1 and do not contain isomorphic copies of c_0 .
Stochastic multi-objective optimization (SMOO) has become increasingly important for balancing competing objectives under uncertainty. However, conventional projection-based methods often incur prohibitive computational costs when the feasible set admits no closed-form projection. This paper proposes the Stochastic Frank–Wolfe (SFW) algorithm, a projection-free framework that leverages linear minimization oracles to bypass expensive projection operations. We investigate the algorithm in a non-oblivious stochastic setting, where the distribution of the random variable depends on the decision variable. By incorporating momentum-based gradient approximation, the proposed method achieves rigorous convergence rate guarantees for both convex and non-convex objectives. In the oblivious stochastic setting, we further establish convergence rates of the algorithm in both expectation and high-probability senses for convex and non-convex cases, respectively. Numerical experiments demonstrate that SFW achieves superior computational efficiency and solution quality compared with existing projection-based approaches.
The primary focus of this work is to provide first-order characterization of convexities for set-valued mappings. We recall the concept of weighted directional derivative for set-valued mappings involving weighted set order relations, and discuss some of its properties. A mean value theorem for set-valued mappings involving weighted directional derivatives is established. We give the definition of convex, quasiconvex, pseudoconvex set-valued mappings involving weighted set order relations, and provide several characterizations of these convexities in terms of weighted directional derivative. We also derive optimality conditions for solutions of the set optimization problems.
