Interior-Point Optimization for Engineering Design: Implementation of the Karmarkar Algorithm in Structural and Water Resource Problems

Although interior-point methods (IPMs) have transformed mathematical programming since 1984, the original projective Karmarkar algorithm is rarely documented step by step on reproducible engineering examples that combine algorithmic transparency with real resource allocation constraints. This article therefore does not propose a new variant of Karmarkar’s algorithm; rather, its scientific contribution is the reproducible MATLAB implementation, canonical-form conversion, and comparative validation of the original projective method against the revised Simplex method and Barnes’ affine scaling variant in two engineering settings. The case studies are (i) the minimum-weight plastic design of a rigid frame with seven candidate plastic hinge locations and six collapse mechanisms and (ii) the optimal allocation of crop patterns in the Caplina Valley (Tacna, Southern Peru), an arid irrigated system with an irrigated command area of 1253 ha, monthly labor availability of 22,239 jornales, and water availability derived from Caplina River discharges at 75% persistence. For Case I, the algorithm reached F = 1.001 in the normalized dual space, which corresponds to F = 4.251 in the original structural objective after applying the scaling factor 17/4; relative to the analytical optimum F* = 4.25, this gives |4.251 − 4.25|/4.25 = 2.4 × 10−4 after 20 iterations. For Case II, the model yielded the maximum net production value of USD 703,135.92, allocating 948.47 ha among 12 crops while satisfying water, labor, market, and land constraints. The double validation confirms the algorithm’s strictly interior trajectory, polynomial-time rationale, and transparent internal parameters (α = 0.7968, ε = 10−8), making the implementation a reproducible benchmark for educational use and for future AI–operations research hybrid solvers in regions with limited access to commercial optimization software.