# Efficient Quadratic Programming Optimization via Staged Computational Procedures: Unconstrained Minimization and Constraint Exploration

## Efficient Quadratic Programming Optimization via Staged Computational Procedures

## DOI:

https://doi.org/10.31258/jomso.v1i2.15## Keywords:

optimal, uncertainty, exploration, programming, quadratic## Abstract

In this paper, we present a computational framework for finding optimal solutions to quadratic programming problems. Our computational process is divided into three steps. Initially, we derive unconstrained minimization of the quadratic programming problem by solving simultaneous equations involving objective function derivatives and confirming its feasibility. Using this discovered point, we identify the violated constraints and direct our search to these specific constraints. The next stage defines the process for determining the unconstrained point on each active constraint violated by the objective function's optimal point. Moving on to the next stage, we use the constraint exploration technique to systematically seek the optimal constrained point at the intersections of two or more violated active constraints as candidates for the optimal solution. The feasibility of the unconstrained point is systematically checked at each level. If the unconstrained point is deemed feasible, then the optimal solution is obtained, and the optimal value of the objective function is found.

## References

Bernau, H. Quadratic programming problems and related linear complementarity problems. 1990, J Optim Theory Appl, Volume 65, pp. 209-222.

Hall, Jonas & Nurkanovic, Armin & Messerer, Florian & Diehl, Moritz., 2021, A Sequential Convex Programming Approach to Solving Quadratic Programs and Optimal Control Problems With Linear Complementarity Constraints. IEEE Control Systems Letters, pp.1-1. 10.1109/LCSYS.2021.3083467.

I. M. Bomze, M. Dür, E. de Klerk, A. Quist, C. Roos, and T. Terlaky, 2000, On copositive programming and standard quadratic optimization problems, J. Global Optim., Volume 18, pp. 301-320.

Ismail Bin Mohd and Yosza bin Dasril, 2000, Constraint exploration method for quadratic programming problem, J. Applied Mathematics and Computation. Volume 112, pp. 161-170.

Fernández-Navarro F, Martínez-Nieto L, Carbonero-Ruz M, Montero-Romero T. 2021, Mean Squared Variance Portfolio: A Mixed-Integer Linear Programming Formulation. Mathematics. Volume 9(3):223.

Kassa, Semu Mitiku , & Tsegay, Teklay Hailay, 2018, An iterative method for trilevel quadratic fractional programming problems using fuzzy goal programming approach Journal of Industrial Engineering International, Volume 14(2), article id. , 264 pp.

R. E. Bellman, L. A. Zadeh, 1970, Decision-Making in a Fuzzy Environment. Management Science Volume 17(4),pp. B-141-B-164.

E. Ammar and H. A Khalifah,2003, Fuzzy portfolio optimization a quadratic programming approach, J. of Chaos, Solitons and Fractals. Volume 18, pp. 1045-1054.

J. J Buckley, E. Esfandiar, and F Thomas, 2002, Fuzzy mathematics in economics and engineering, Physica-Verlag (2002)

Petr Ya. Ekel, Marden Menezes, Fernando H. Schuffner Neto, 2007, Decision making in a fuzzy environment and its application to multicriteria power engineering problems, Nonlinear Analysis: Hybrid Systems, Volume 1(4), pp. 527-536