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All Journal Griya Journal of Mathematics Education and Application
Aprilia Rahmawati
Universitas Sriwijaya
Education Mathematics

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Teori Dualitas Lagrange dalam Optimasi Non-Linier: Tinjauan Naratif dari Perspektif Analisis Geometris Modern Aprilia Rahmawati; Sisca Octarina; Fitri Maya Puspita
Griya Journal of Mathematics Education and Application Vol. 6 No. 2 (2026): Juni 2026
Publisher : Pendidikan Matematika FKIP Universitas Mataram

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.29303/griya.v6i2.1044

Abstract

Lagrangian duality theory constitutes a fundamental framework in nonlinear optimization analysis, connecting primal problems with dual problems through the Lagrange function. This article presents a comprehensive narrative review of Lagrangian duality theory development from the perspective of modern geometric analysis. We explore the geometric structure of primal-dual spaces, Karush-Kuhn-Tucker (KKT) optimality conditions, and strong and weak duality theorems in the context of constrained optimization. The discussion encompasses geometric interpretations of saddle points, convexity in duality theory, and the duality gap as a solution quality measure. Furthermore, we analyze theoretical applications from functional analysis and topology perspectives, including the role of reflexive Banach spaces and Fréchet differentiability in characterizing optimal solutions. The findings demonstrate that geometric approaches provide profound insights into optimization problem structures and open new perspectives in numerical algorithm development. This review contributes to a more robust theoretical understanding of the mathematical foundations of nonlinear optimization and its relevance in contemporary applied mathematics.
Optimasi Penjadwalan Produksi Multi-Mesin Menggunakan Model Integer Programming pada UKM Manufaktur Aprilia Rahmawati; Fitri Maya Puspita; Sisca Octarina
Griya Journal of Mathematics Education and Application Vol. 6 No. 3 (2026): September 2026
Publisher : Pendidikan Matematika FKIP Universitas Mataram

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.29303/griya.v6i3.1430

Abstract

Integer programming (IP) constitutes a fundamental framework in combinatorial optimization, extending linear programming by requiring decision variables to take integer values. This article presents a comprehensive narrative review of integer programming theory from the perspective of polyhedral analysis and cutting-plane methods. We examine the geometric relationship between the linear programming relaxation and the integer hull, the role of total unimodularity in guaranteeing integral extreme points, and the structural properties of valid inequalities that tighten this relaxation. The discussion covers the derivation of Chvátal-Gomory cuts, the geometry of branch-and-bound as a recursive partitioning of the feasible region, and Lagrangian relaxation as a dual bounding technique for integer programs, including the resulting integrality gap that distinguishes integer from linear duality. We further discuss recent developments connecting polyhedral theory with machine-learning-based cutting-plane selection. The findings show that polyhedral and geometric perspectives provide a coherent theoretical foundation for understanding why integer programs are computationally harder than their linear relaxations and how modern solvers exploit geometric structure to close the integrality gap. This review contributes to a deeper theoretical understanding of integer programming as a cornerstone of discrete optimization.
Co-Authors Fitri Maya Puspita Sisca Octarina