A geometric reformulation of the bilevel parameter optimization problem to a single level non-linear programming problem with applications to phase equilibria

Multi-parametric approach for nonlinear bilevel optimization problems and beyond

One of the peculiar features of multi-parametric approach for bilevel programs is that most methods using this approach can be extended to tri-level (and generally to $k$-level) programs, which is not always the case with other non-heuristic solution methods. However, most of existing multi-parametric methods work well when the constraint of the lower-level problem is polyhedral. In this article we propose a multi-parametric programming based solution algorithm for bilevel optimization problems whose lower-level problem involves convex smooth nonlinear constraints. The method is also extended to solve some classes of $k$-level convex optimization problems with nonlinear constraints. The algorithm recasts the lower-level problem as a multi-parametric problem and employs an equivalent barrier problem reformulation. The solution obtained through multi-parametric programming is incorporated in the upper-level problem to create a set of single-level optimization problems which are solved using standard global optimization techniques. The proposed algorithm can give an exact global solution to some class of nonlinear Multi-level problems with convex nonlinear constraints.

A significant biomimetic numerical optimization approach for solving complex Non-linear problems

One of the hard problems in the modern world of high-level science and engineering is to find the solution of non-linear equations. A wide range of real-life applications such as climate prediction, chemical reactions, physical phenomenon, spacecraft trajectory determination, finance and economic modeling etc. exhibits non-linear dynamics in nature. A new numerical optimization solver has been proposed in this paper. The suggested method is an algorithm that combines numerical iterative method with meta-heuristic algorithms for tackling intricate Non-linear system of equations (NSE). The techniques that are used in constructing the proposed method include; Newton fourth order variation and Coati Optimization Algorithm (COA). The COA is used in order to conduct an effective global exploration of the solution space, and the fourth-order iterative approach helps to achieve quick and precise convergence at the local search stage. This combination of methods results in the increased efficiency of the computation process and accuracy of the solution process. The stability, reliability and efficiency of the proposed approach has been verified by the applicability to large and complex non-linear system of equations, which contain optimization benchmark functions and a single real-life application of Economic modeling. Comparative analyses are thoroughly done against a number of state-of-the-art metaheuristic algorithms, which include the Slime Mould Optimization Algorithm (SMA),  Pelican Optimization Algorithm (POA), Walrus Optimization Algorithm (WOA) and the original COA. The findings of the numerical simulations show that the proposed strategy always demonstrates high performance in a variety of measures, including lower computational time, fewer converge iterations, and better fitness function values. This parametric study of the suggested method makes it a prospective instrument in terms of the optimization of complex systems and the accuracy of the solutions.

SEMI-ANALYTICAL APPROXIMATION OF THE SET PARTITION PROBLEM USING THE HOMOTOPY ANALYSIS METHOD

The Set Partition Problem (SPP) is a classical NP-hard combinatorial optimization problem with applications in scheduling, resource allocation, cryptography, and operations research. In this work, a semi-analytical framework based on the Homotopy Analysis Method (HAM) is developed for approximating solutions of the SPP. The discrete partitioning problem is first reformulated as a constrained continuous optimization problem using Lagrange multipliers. A homotopy is then constructed between an initial guess and the full nonlinear system, and a Padé $[1, 1]$ series representation is employed to obtain approximate analytical solutions. The resulting algebraic equations are solved up to second order in the homotopy parameter, and the explicit Taylor-series derivation of these equations is given, together with a discussion of the conditions under which the resulting linear system for the Padé coefficients becomes degenerate. A numerical example involving a six-element set is investigated, and the influence of the convergence-control parameter on a constraint-violation residual is analyzed statistically over multiple random initial guesses, using a statistically motivated outlier criterion. The results show that the residual varies non-monotonically with the convergence-control parameter, with no clearly defined convergence region, and that the rounded partition assignments obtained from the second-order Padé approximation fail to satisfy the exact partition condition for the tested samples. The study illustrates the potential and, in its present second-order form, the significant limitations of semi-analytical homotopy techniques when applied to NP-hard combinatorial optimization problems. Furthermore, it identifies higher-order approximations and pole-avoidance strategies as necessary directions for improvement.

Parareal Algorithm For Gradient Flow Arising From Orthogonality-Constrained Matrix Optimization Problems

A new dual-population constrained multi-objective evolutionary optimization algorithm with repair constraint handling for structural optimization

This study introduces a novel constrained multi-objective evolutionary algorithm, termed DPCME, which employs two interacting populations that exchange information, enabling effective global exploration and reducing the risk of convergence to local optima.

Constrained Optimization using an Evolutionary - Augmented Lagrangian approach

This work proposes a Hybrid Augmented Lagrangian (HyAL) method that combines the constraint-handling capabilities of the AL framework with the exploratory power of population-based search, and demonstrates that HyAL effectively solves constrained optimization problems, including high-dimensional cases where population-based methods struggle.