On the Finite Complexity of Solutions in a Degenerate System of Quadratic Equations: Exact Formula.

The paper describes an application of the p-regularity theory to Quadratic Programming (QP) and nonlinear equations with quadratic mappings. In the first part of the paper, a special structure of the nonlinear equation and a construction of the 2-factor operator are used to obtain an exact formula for a solution to the nonlinear equation. In the second part of the paper, the QP problem is reduced to a system of linear equations using the 2-factor operator. The solution to this system represents a local minimizer of the QP problem along with its corresponding Lagrange multiplier. An explicit formula for the solution of the linear system is provided. Additionally, the paper outlines a procedure for identifying active constraints, which plays a crucial role in constructing the linear system.

On the Locally Polynomial Complexity of the Projection-Gradient Method for Solving Piecewise Quadratic Optimisation Problems.

This paper proposes a method for solving optimisation problems involving piecewise quadratic functions. The method provides a solution in a finite number of iterations, and the computational complexity of the proposed method is locally polynomial of the problem dimension, i.e., if the initial point belongs to the sufficiently small neighbourhood of the solution set. Proposed method could be applied for solving large systems of linear inequalities.

Autologous Matrix-Induced Chondrogenesis (AMIC) for Focal Chondral Lesions of the Knee: A 2-Year Follow-Up of Clinical, Proprioceptive, and Isokinetic Evaluation.

(1) Background: The autologous matrix-induced chondrogenesis (AMIC) is a bio-orthopedic treatment for articular cartilage damage. It combines microfracture surgery with the application of a collagen membrane. The aim of the present study was to report a medium-term follow-up of patients treated with AMIC for focal chondral lesions. (2) Methods: Fourty-eight patients treated surgically and 21 control participants were enrolled in the study. To evaluate the functional outcomes, the proprioceptive (postural stability, postural priority) and isokinetic (peak value of maximum knee extensor and flexor torque in relation to body mass and the total work) measurements were performed. To evaluate the clinical outcomes, the Lysholm score and the IKDC score were imposed. (3) Results: Compared to the preoperative values, there was significant improvement in the first 2 years after intervention in the functional as well as subjective outcome measures. (4) Conclusions: AMIC showed durable results in aligned knees.