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Triloki Nath

Publications and source records attributed to Triloki Nath.

8 recordsLinked to original sources
arXiv2026

A Generalized Heron-Waist Problem: Optimality Conditions and Convergence Analysis

This paper introduces and solves the Generalized Heron-Waist Problem (GHWP), that integrates the classical Heron problem of optimal hub location and the waist problem of minimal-perimeter configuration. The GHWP seeks an optimal closed polygonal chain with weights whose vertices are constrained to lie in the given nonempty, closed, and convex sets, while simultaneously minimizing weighted distances to a central hub point. This coupled formulation naturally models systems in which cyclic internal connectivity and radial access to a hub must be optimized jointly a structural feature that arises in applications such as supply-chain design, transportation planning, and communication infrastructures. Using modern convex analysis tools, we establish existence of optimal solutions under boundedness and general position assumptions of sets, we prove uniqueness when constraint sets are strictly convex with positive weights. We also derive first order necessary and sufficient optimality conditions using subdifferential calculus. For computation, we develop a Projected Subgradient Algorithm (PSA) and we prove convergence of the best-iterate sequence under classical diminishing step size rules. Numerical illustrations in $\mathbb{R}^2$ and $\mathbb{R}^3$ are provided to validate the algorithm's robustness across diverse geometries and weighting schemes.

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arXiv2025

Minimal Perimeter Triangle in Nonconvex Quadrangle: Generalized Fagnano Problem

In 1775, Fagnano introduced the following geometric optimization problem: inscribe a triangle of minimal perimeter in a given acute-angled triangle. A widely accessible solution is provided by the Hungarian mathematician L. Fejer in 1900. This paper presents a specific generalization of the classical Fagnano problem, which states that given a nonconvex quadrangle (having one reflex angle and others are acute angles), find a triangle of minimal perimeter with exactly one vertex on each of the sides that do not form reflex angle, and the third vertex lies on either of the sides forming the reflex angle. We provide its geometric solution. Additionally, we establish an upper bound for the classical Fagnano problem, demonstrating that the minimal perimeter of the triangle inscribed in a given acute-angled triangle cannot exceed twice the length of any of its sides

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arXiv2025

A Generalized Waist Problem: Optimality Condition and Algorithm

Many years ago John Tyrell a lecturer at King's college London challenged his Ph.D. students with the following puzzle: show that there is a unique triangle of minimal perimeter with exactly one vertex to lie on one of three given lines, pairwise disjoint and not all parallel in the space. The problem in literature is known as the waist problem, and only convexity rescued in this case. Motivated by this we generalize it by replacing lines with a number of convex sets in the Euclidean space and ask to minimize the sum of distances connecting the sets by means of closed polygonal curve. This generalized problem significantly broadens its geometric and practical scope in view of modern convex analysis. We establish the existence of solutions and prove its uniqueness under the condition that at least one of the convex sets is strictly convex and all are in general position: each set can be separated by convex hull of others. A complete set of necessary and sufficient optimality conditions is derived, and their geometric interpretations are explored to link these conditions with classical principles such as the reflection law of light. To address this problem computationally, we develop a projected subgradient descent method and prove its convergence. Our algorithm is supported by detailed numerical experiments, particularly in cases involving discs and spheres. Additionally, we present a real-world analogy of the problem in the form of inter-island connectivity, illustrating its practical relevance. This work not only advances the theory of geometric optimization but also contributes effective methods and insights applicable to facility location, network design, robotics., computational geometry, and spatial planning.

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arXiv2025

A Generalized $(k,m)$ Heron Problem:Optimality Conditions and Algorithm

This paper presents a new extension of the classical Heron problem, termed the generalized $(k,m)$-Heron problem, which seeks an optimal configuration among $k$ feasible and $m$ target non-empty closed convex sets in $\mathbb{R}^n$. The problem is formulated as finding a point in each set that minimizes the pairwise distances from the points in the $k$-feasible sets to the points in the $m$-target sets. This formulation leads to a convex optimization framework that generalizes several well-known geometric distance problems. Using tools from convex analysis, we establish fundamental results on existence, uniqueness, and first-order optimality conditions through subdifferential calculus and normal cone theory. Building on these insights, a Projected Subgradient Algorithm (PSA) is proposed for numerical solution, and its convergence is rigorously proved under a diminishing step-size rule. Numerical experiments in $\mathbb{R}^2$ and $\mathbb{R}^3$ illustrate the algorithm's stability, geometric accuracy, and computational efficiency. Overall, this work provides a comprehensive analytical and algorithmic framework for multi-set geometric optimization with promising implications for location science, robotics, and computational geometry.

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arXiv2019

Some remarks on weak generalizations of minima and quasi efficiency

In this note, we remark, with sufficient mathematical rigor, that many weak generalizations of the usual minimum available in the literature are not true generalizations. Motivated by the Ekeland Variational Principle, we provide, first time, the criteria for weaker generalizations of the usual minimum. Further, we show that the quasi efficiency, recently used in Bhatia et al. (Optim. Lett. 7, 127-135 (2013)) and introduced in Gupta et al. ( Bull. Aust. Math. Soc. 74, 207-218 (2006)) is not a true generalization of the usual efficiency. Since the former paper relies heavily on the results of later one, so we discuss the later paper. We show that the necessary optimality condition is a consequence of the local Lipschitzness and sufficiency result is trivial in the later paper. Consequently, the duality results of the same paper are also inconsistent.

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arXiv2018

Enhanced Fritz John Stationarity, New Constraint Qualifications and Local Error Bound for Mathematical Programs with Vanishing Constraints

In this paper, we study the difficult class of optimization problems called the mathematical programs with vanishing constraints or MPVC. Extensive research has been done for MPVC regarding stationary conditions and constraint qualifications using geometric approaches. We use the Fritz John approach for MPVC to derive the M-stationary conditions under weak constraint qualifications. An enhanced Fritz John type stationary condition is also derived for MPVC, which provides the notion of enhanced M-stationarity under a new and weaker constraint qualification: MPVC-generalized quasinormality. We show that this new constraint qualification is even weaker than MPVC-CPLD. A local error bound result is also established under MPVC-generalized quasinormality.

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arXiv2015

Differentiability of Distance Function and The Proximinal Condition implying Convexity

We establish a necessary and sufficient condition for the differentiability of the distance function generated by a nonempty closed set K in a real normed linear space X under a proximinality condition on K. We do not assume the uniform differentiability constraints on the norm of the space as in Giles [16]. Hence, our result advances that of Giles [16]. We prove that the proximinal condition of Giles [16] is true for almost suns. The proximinal condition ensures convexity of an almost sun in some class of strongly smooth spaces under a differentiability condition of the distance function. A necessary and sufficient condition is obtained for the convexity of Chebyshev sets in Banach spaces with rotund dual.

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