arXiv subjects

Rachit Mehra

Publications and source records attributed to Rachit Mehra.

4 recordsLinked to original sources

arXiv2026

Reciprocal-Manifold Annealed KKT Flows for Constrained Optimization: Application to the Nonconvex AC Optimal Power Flow

Safety-critical optimization applications, such as real-time power system operation, maintain feasibility at every intermediate step, not merely at convergence. Existing approaches either violate constraints mid-solve (interior-point methods) or enforce feasibility through per-instant quadratic programming subproblems with cubic computational cost and unbounded worst-case execution time. We propose a continuous-time optimization framework for smooth constrained nonlinear problems that preserves feasibility throughout the optimization process without requiring projection operators, quadratic programming subproblems, or other per-iteration optimization routines. The method is built around a reciprocal multiplier manifold, which establishes an explicit relationship between inequality constraints and their associated Lagrange multipliers. By designing a continuous multiplier update law, the manifold is shown to remain forward invariant, while the resulting dynamics are equivalent to continuous-time logarithmic barrier gradient descent. The proposed framework naturally extends to multiple inequality constraints, equality constraints, nonconvex feasible sets, and infeasible initial conditions. The method is further enhanced through an augmented Uzawa flow that eliminates oscillatory transients commonly observed in classical primal-dual saddle-point dynamics. The effectiveness of the proposed approach is applied to the AC Optimal Power Flow problem of IEEE 9-bus and IEEE 57-bus systems. Numerical results show convergence to solutions within 0.4\% of the benchmark optimum while maintaining strict feasibility of all constraints. A computational complexity analysis shows that the proposed dynamics reduce the per-step computational cost from cubic to linear complexity. Finally, dynamic tracking studies under time-varying operating conditions demonstrate reliable feasibility preservation.

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arXiv2026

On the Nesterov's acceleration: A NAIM perspective

We present a unifying Nearly Asymptotically Invariant Manifold (NAIM) framework for understanding Nesterovs Accelerated Gradient (NAG) method. By lifting the first-order gradient flow into a second-order phase space we construct a NAIM a slow, attracting graph and show that acceleration emerges from a curvature aware perturbation of this graph. The evolving slope of the perturbed manifold is governed by a Differential Riccati Equation (DRE), which enforces strict tangency of the vector field to the manifold surface. In the quadratic case the DRE reduces to an Algebraic Riccati Equation (ARE), and the requirement of spectral resonance equal contraction rates across all curvature modes uniquely determines the damping coefficient, directly yielding the continuous time Nesterov ODE. Fenichels theorem then extends this picture rigorously to general smooth, strongly convex landscapes: normal hyperbolicity guarantees persistence of the accelerated manifold despite varying Hessian curvature. The method is further extended to unified geometric derivation of NAG methods for smooth convex and strongly convex optimization in the discrete case. We exploit the underlying geometric structure and derive both cases from the same principle of preserving the projective structure under discretization process. A Lie Trotter splitting separates the linear dissipative dynamics from the nonlinear gradient flow. The dissipative subsystem is integrated by the Cayley (bilinear) transform, which preserves the underlying projective (Mobius) structure unconditionally and produces the classical Nesterov momentum coefficient as the unique Pade multiplier. For the convex case, projective flatness (vanishing Schwarzian derivative) uniquely selects the time-varying damping recovering the canonical Nesterov ODE for convex functions.

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arXiv2025

On the Dynamics of Acceleration in First order Gradient Methods

Ever since the original algorithm by Nesterov (1983), the true nature of the acceleration phenomenon has remained elusive, with various interpretations of why the method is actually faster. The diagnosis of the algorithm through the lens of Ordinary Differential Equations (ODEs) and the corresponding dynamical system formulation to explain the underlying dynamics has a rich history. In the literature, the ODEs that explain algorithms are typically derived by considering the limiting case of the algorithm maps themselves, that is, an ODE formulation follows the development of an algorithm. This obfuscates the underlying higher order principles and thus provides little evidence of the working of the algorithm. Such has been the case with Nesterov algorithm and the various analogies used to describe the acceleration phenomena, viz, momentum associated with the rolling of a Heavy-Ball down a slope, Hessian damping etc. The main focus of our work is to ideate the genesis of the Nesterov algorithm from the viewpoint of dynamical systems leading to demystifying the mathematical rigour behind the algorithm. Instead of reverse engineering ODEs from discrete algorithms, this work explores tools from the recently developed control paradigm titled Passivity and Immersion approach and the Geometric Singular Perturbation theory which are applied to arrive at the formulation of a dynamical system that explains and models the acceleration phenomena. This perspective helps to gain insights into the various terms present and the sequence of steps used in Nesterovs accelerated algorithm for the smooth strongly convex and the convex case. The framework can also be extended to derive the acceleration achieved using the triple momentum method and provides justifications for the non-convergence to the optimal solution in the Heavy-Ball method.

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arXiv2025

On the Dynamics of Control

We present a dynamical system approach for the control of a nonlinear dynamical system by defining the control problem in a Fiber bundle framework. The constructive procedure derived results in the generation of a NHIM/NAIM which facilitates the use of tools and ideas from dynamical system theory to analyze and understand the properties associated with the controlled system. The time scale separation, decoupling of system dynamics, and their role in the system behavior are analyzed. An overview of the benefits of the above approach is demonstrated by briefly discussing three main application areas.

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