Distributed Time-Varying Constrained Convex Optimization: Finite-Time/Fixed-Time Convergence

This article investigates a distributed time-varying optimization problem with inequality constraints, aiming to find finite-time and fixed-time convergent solutions free from initialization. A nonsmooth optimization algorithm for state consensus achieving within a finite or fixed time is presented by designing a projection-based log-barrier penalty cost function to meet the constraints and introducing integral sliding mode subsystems to guarantee zero-gradient-sum. With the use of the projection idea, the penalized functions are always well defined (i.e., satisfying the logarithmic definition) for any system states, which avoids initializing of certain parameters. An adaptive gain scheme without any extra global information is presented. The time-varying zero-gradient-sum method here is feasible for cost functions with nonidentical Hessian matrixes, and applicable to finite-time or fixed-time optimal consensus tracking. The effectiveness and superiority of our algorithms are verified with numerical simulations.