This paper investigates the necessary optimality conditions for a multiobjective bilevel optimization problem in which both the upper- and lower-level objective functions are interval-valued. By reformulating the hierarchical model into a single-level problem through optimal value reformulation, we derive Karush-Kuhn-Tucker-type optimality conditions under an appropriate nonsmooth Abadie-type constraint qualification, expressed in terms of approximations. Examples are provided to demonstrate the applicability of our findings and to underscore the limitations of certain previously published results.

We investigate a variant of the class of prox-convex functions introduced in Grad and Lara [An extension of the proximal point algorithm beyond convexity. J Glob Optim. 2022;82:313–329] to the context of Banach spaces, namely prox-convexity and generalized prox-convexity, respectively. The proximity operators of such a function is single-valued, generalized firmly nonexpansive and weakly continuous. The Moreau envelope of such a function is directionly differentiable. We show that the classical proximal point algorithm remains convergent when the convexity of the proper lower semicontinuous function to be minimized is relaxed to prox-convexity or generalized prox-convexity. We also proved the convergence of the function values; the asymptotic rate n − 1 reaches its optimal value.

As physics provides the equations of motion of a body, this paper formulates, for the first time, at the conceptual/mathematical levels, the inequations of motion of an individual seeking to meet his needs/desires in an adaptive/flexible way. Successful (failed) dynamics perform a succession of moves that are, at once, satisficing (improving enough) and worthwhile (free from too many sacrifices), or not. They approach or reach needs and desires (fall into traps). They balance the desired speed of approach to a desired end (a distal promotion goal) with the size of the required immediate sacrifices to go fast (a proximal prevention goal). Therefore, each period, need/desire satisfaction success requires enough self-control to be able to make, in the long run, sufficient progress in need/desire satisfaction without enduring, in the short run, too many sacrifices. A simple example (lose or gain weight) shows that the size of successful moves must be neither too small nor too long. To save space, an extended version of this paper solves this problem by using an approximate gradient algorithm. This paper opens the door to ‘algorithmic psychology’ using the recent variationality approach of stay/change human dynamics.

In this article, we aim to approximate a solution to the bilevel equilibrium problem (BEP) for short: find x ¯ ∈ S f such that g ( x ¯ , y ) ≥ 0 , ∀ y ∈ S f , where S f = { u ∈ K : f ( u , z ) ≥ 0 , ∀ z ∈ K } . Here, K is a closed convex subset of a real Hilbert space H , and f and g are two real-valued bifunctions defined on K × K . We propose an inertial version of the proximal splitting algorithm introduced by Z. Chbani and H. Riahi: [Weak and strong convergence of prox-penalization and splitting algorithms for bilevel equilibrium problems. Numer. Algebra Control Optim. 2013;3:353–366]. Under suitable conditions, we establish the weak and strong convergence of the sequence generated by the proposed iterative method. We also report a numerical example illustrating our theoretical result.

In 1956, Frank and Wolfe proved that a quadratic function which is bounded from below on a nonempty polyhedral convex set attains its infimum there. In this paper, we propose sufficient conditions for the solution of nonconvex polynomial programming problems via a Frank–Wolfe-type theorem. We also provide several numerical examples to illustrate the results. The last section presents an application through an Eaves type theorem and refines the main results for quadratic programming problems.

This study introduces a mathematical framework to investigate the viability and reachability of production systems under constraints. We develop a model that incorporates key decision variables, such as pricing policy, quality investment, and advertising, to analyze short-term tactical decisions and long-term strategic outcomes. In the short term, we constructed a capture basin that defined the initial conditions under which production viability constraints were satisfied within the target zone. In the long term, we explore the dynamics of product quality and market demand to achieve and sustain the desired target. The Hamilton-Jacobi-Bellman (HJB) theory characterizes the capture basin and viability kernel using viscosity solutions of the HJB equation. This approach, which avoids controllability assumptions, is well suited to viability problems with specified targets. It provides managers with insights into maintaining production and inventory levels within viable ranges while considering product quality and evolving market demand. We numerically studied the HJB equation to design and test computational methods that validate the theoretical insights. Simulations offer practical tools for decision-makers to address operational challenges while aligning with the long-term sustainability goals. This study enhances the production system performance and resilience by linking rigorous mathematics with actionable solutions.

Proximal point algorithm has found many applications, and it has been playing fundamental roles in the understanding, design, and analysis of many first-order methods. In this paper, we derive the tight convergence rate in subgradient norm of the proximal point algorithm, which was conjectured by Taylor, Hendrickx and Glineur [SIAM J. Optim. 2017:27;1283–1313]. This sort of convergence results in terms of the residual (sub)gradient norm is particularly interesting when considering dual methods, where the dual residual gradient norm corresponds to the primal distance to feasibility.

We introduce and study the notion of ( e , y ) -conjugate for a proper and e-convex function in locally convex spaces, which is an extension of the concept of the conjugate. The mutual relationships between the concepts of ( e , y ) -conjugacy and e-subdifferential are presented. Moreover, some applications of these notions in optimization are established.

A plethora of models from real-world applications can boil down to the composite optimization. In this paper, we focus on the composite optimization whose objective is the precomposition of convex function with nonlinear mapping. By recourse to the conjugacy, we formulate the composite optimization as a saddle point problem with nonlinear function (‘nonlinear SPP’). By pursuing the track of nonlinear primal–dual hybrid gradient method in the seminal work (Valkonen, Inverse Problems, 30, 2014). We propose a variant nonlinear PDHG with variable metric and provide an alternative rationale for the convergence analysis by leveraging solution mapping and localization in variational analysis. Furthermore, the proposed method can be applicable to the nonlinear SPP with generic convex objectives.

In this paper, we first establish several versions of generalized Farkas lemmas for robust linear systems with adjustable variables. These results are obtained by transforming the adjustable robust linear system, via a specialized technique, into a more complex robust system to which existing tools and techniques from the context of robust Farkas lemmas can be applied. The results just derived are then employed as the main tools to study the adjustable two-stage robust linear programe with a general uncertainty set, referred to briefly as the adjustable robust linear problem. For this problem, we propose two distinct formulations of dual problems and establish several optimality conditions, together with strong duality results corresponding to the associated primal–dual pairs. A special case, namely the adjustable robust linear problem with fixed recourse (i.e. where the recourse matrix is independent of the uncertainty parameter), is examined in detail. For this case, we show that one of its dual problems is a convex one that, in many instances, can be efficiently solved using well-known optimization packages. Several examples are provided throughout the paper to illustrate the theoretical results.