Optimization Theory: Concepts, Methods, and Applications

Introduced by Polyak in 1966, the class of strongly quasiconvex functions includes some interesting nonconvex members, like the square root of the Euclidean norm or ratios with a nonnegative strongly convex numerator and a concave and positive denominator. In this talk, we survey the most relevant examples of strongly quasiconvex functions and results involving them available in the literature at the moment. In particular, we recall some recent algorithms for minimizing such functions, and hint toward some directions where additional investigations would be welcome. The talk is based on joint work with Felipe Lara, Raúl Marcavillaca, and Huu-Nhan Nguyen.

Providing the existence of an intersection point for a family of sets is commonly useful in many problems. Various studies have been done in this regard within the framework of KKM theory. The concept of weak KKM has been applied in some papers within this objective. In these results, the existence of a solution is often linked to certain closedness and compactness conditions that are not satisfied in many real applications. On the other hand, various notions of pseudomonotonicity are applied to variational inequalities, optimization and other problems in nonlinear analysis. Here, first we introduce a new concept of topological pseudomonotonicity for a pair of set-valued maps. Then, applying this condition, we present an intersection theorem under relaxed closedness and coercivity conditions. Moreover, by introducing new concept of upper continuity for bifunctions, some applications of this result will be presented.

In this paper, we introduce and study a Halpern inertial method for solving the general
perturbed Krasnoselskii–Mann type algorithm in Hilbert space settings, where the underlying mapping is
quasi–nonexpansive. We discuss convergence analysis of the method under some mild assumptions on the
control sequences. We additionally present a numerical example to demonstrate the effectiveness of the
method. Finally, we apply the method in solving image restoration problems. We evaluate the quality
of the Improvement in the Signal–to–Noise Ratio (ISNR) and the restored images using the Structural
Similarity Index Measure (SSIM) metrics. The results of this work generalize and expand upon numerous
results found in the literature.

This work presents a new iterative scheme and establishes its convergence results to approximate the fixed points of nonexpansive mapping. In particular, we demonstrate effectiveness of our proposed iterative scheme in the image restoration process by formulating the problem as a split feasibility problem (SFP). A comparative analysis reveals that our scheme not only converges faster than some classical iterative processes but also achieves good restoration quality, thereby bridging the gap between abstract convergence results and real-world computational performance. The integration of fixed point methods with modern image restoration highlights the novelty of our approach and underscores its potential as a powerful tool for advancing variational and projection-based techniques in imaging sciences.