Alexander Y. Kruger

We develop a methodology for closing duality gap and guaranteeing strong duality in infinite convex optimization. Specifically, we examine two new Lagrangian-type dual formulations involving infinitely many dual variables and infinite sums of functions. Unlike the classical Haar duality scheme, these dual problems provide zero duality gap and are solvable under the standard Slater condition. We derive general optimality conditions/multiplier rules by applying subdifferential rules for infinite sums established in Hantoute, A J. Optim. Theory Appl. 2026, https://doi.org/10.1007/s10957-025-02916-w ,

The paper extends the widely used in optimisation theory decoupling techniques to infinite collections of functions. Extended concepts of uniform lower semicontinuity and firm uniform lower semicontinuity are discussed. The main theorems give fuzzy subdifferential necessary conditions (multiplier rules) for a local minimum of the sum of an infinite collection of functions and fuzzy subdifferential sum rules without the traditional Lipschitz continuity assumptions. More subtle “quasi” versions of the uniform infimum and uniform lower semicontinuity properties are also discussed.

We develop a methodology for closing duality gap and guaranteeing strong duality in infinite convex optimization. Specifically, we examine two new Lagrangian-type dual formulations involving infinitely many dual variables and infinite sums of functions. Unlike the classical Haar duality scheme, these dual problems provide zero duality gap and are solvable under the standard Slater condition. Then we derive general optimality conditions/multiplier rules by applying subdifferential rules for infinite sums established in [13].

In this paper, we study the finite-time convergence of the time-varying dynamical systems for solving convex and nonconvex optimization problems in different scenarios. We first show the asymptotic convergence of the trajectories of dynamical systems while only requiring convexity of the objective function. Under the Kurdyka–Łojasiewicz (KL) exponent of the objective function, we establish the finite-time convergence of the trajectories to the optima from any initial point. Making use of the Moreau envelope, we adapt our finite-time convergent algorithm to solve weakly convex nonsmooth optimization problems. In addition, we unify and extend the contemporary results on the KL exponent of the Moreau envelope of weakly convex functions. A dynamical system is also introduced to find a fixed point of a nonexpansive operator in finite time and fixed time under additional regularity properties. We then apply it to address the composite optimization problems with finite-time and fixed-time convergence.