Abstract This paper investigates a path-following method inspired by the semismooth ∗ approach for solving algebraic inclusions, with a primary emphasis on the role of strong subregularity and its crucial role in ensuring the robustness and stability of the path-following method, as it provides a framework to uniformly control the distance between the input and the solution set across a continuous path. We explore the problem of finding a mapping $x: [0, T] \longrightarrow \mathbb{R}^{n} $ x : [ 0 , T ] ⟶ R n that satisfies $0 \in F(t, x(t)) $ 0 ∈ F ( t , x ( t ) ) for each $t \in [0, T] $ t ∈ [ 0 , T ] , where $F $ F is a set-valued mapping from $\mathbb{R} \times \mathbb{R}^{n} $ R × R n to $\mathbb{R}^{n} $ R n . We consider an approach based on mappings exhibiting uniform semismooth ∗ properties along continuous trajectories, thereby maintaining a uniform grid error throughout the interval. Error estimates demonstrate an $o(h)$ o ( h ) grid error with respect to the discretization parameter under natural regularity assumptions. Numerical experiments applied to electric circuit and elastoplastic models confirm the efficiency and robustness of the method.

Abstract Metric regularity is among the central concepts of nonlinear and variational analysis, constrained optimization, and their numerous applications. However, metric regularity can be elusive for some important ill-posed classes of problems including polynomial equations, parametric variational systems, smooth reformulations of complementarity systems with degenerate solutions, etc. The study of stability issues for such problems can often not rely on the machinery of first-order variational analysis, and so higher-order regularity concepts have been proposed in recent years. In this paper, we investigate some notions of mixed-order regularity by using advanced tools of first-order and second-order variational analysis and generalized differentiation of both primal and dual types. Efficient characterizations of such mixed-order regularity concepts are established by employing a fresh notion of the least singular value function. The obtained conditions are applied to deriving constructive criteria for mixed-order regularity in coupled constraint and variational systems.

The derivation of necessary conditions of optimality for optimal control problems with state/control constraints is a challenging task, not always clearly understood. In this paper, we present a didactic approach based on the method of exponential penalty functions allowing, in our setting, to significantly simplify the proofs. This work may be viewed as a proposal of a course designed for students familiar with the basics of optimal control theory and elements of functional analysis. To make the analysis as simple as possible, we concentrate on the case of optimal control problem with fixed initial point and free end state.

When the lower-level optimal solution set-valued mapping of a bilevel optimization problem is not single-valued, we are faced with an ill-posed problem, which gives rise to the optimistic and pessimistic bilevel optimization problems, as tractable algorithmic frameworks. However, solving the pessimistic bilevel optimization problem is far more challenging than the optimistic one; hence, the literature has mostly been dedicated to the latter class of the problem. The Scholtes relaxation has appeared to be one of the simplest and most efficient ways to solve the optimistic bilevel optimization problem in its Karush-Kuhn-Tucker (KKT) reformulation or the corresponding more general mathematical program with complementarity constraints (MPCC). Inspired by such a success, this paper studies the potential of the Scholtes relaxation in the context of the pessimistic bilevel optimization problem. To proceed, we consider a pessimistic bilevel optimization problem, where all the functions involved are at least continuously differentiable. Then assuming that the lower-level problem is convex, the KKT reformulation of the problem is considered under the Slater constraint qualification. Based on this KKT reformulation, we introduce the corresponding version of the Scholtes relaxation algorithm. We then construct theoretical results ensuring that the limit of a sequence of global/local optimal solutions (resp. stationary points) of the aforementioned Scholtes relaxation is a global/local optimal solution (resp. stationary point) of the KKT reformulation of the pessimistic bilevel program. The results are accompanied by technical constructions ensuring that the Scholtes relaxation algorithm is well-defined or that the corresponding parametric optimization problem is more tractable. Furthermore, we perform some numerical experiments to assess the performance of the Scholtes relaxation algorithm using various examples. In particular, we study the effectiveness of the algorithm in obtaining solutions that can satisfy the corresponding C-stationarity concept.