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Andrew Eberhard

Publications and source records attributed to Andrew Eberhard.

17 recordsLinked to original sources
arXiv2026

Fast Gradient Algorithm with Dry-like Friction and Nonmonotone Line Search for Nonconvex Optimization Problems

In this paper, we propose a fast gradient algorithm for the problem of minimizing a differentiable (possibly nonconvex) function in Hilbert spaces. We first extend the dry friction property for convex functions to what we call the dry-like friction property in a nonconvex setting, and then employ a line search technique to adaptively update parameters at each iteration. Depending on the choice of parameters, the proposed algorithm exhibits subsequential convergence to a critical point or full sequential convergence to an ``approximate'' critical point of the objective function. We also establish the full sequential convergence to a critical point under the Kurdyka--Lojasiewicz (KL) property of a merit function. Thanks to the parameters' flexibility, our algorithm can reduce to a number of existing inertial gradient algorithms with Hessian damping and dry friction. By exploiting variational properties of the Moreau envelope, the proposed algorithm is adapted to address weakly convex nonsmooth optimization problems. In particular, we extend the result on KL exponent for the Moreau envelope of a convex KL function to a broad class of KL functions that are not necessarily convex nor continuous. Simulation results illustrate the efficiency of our algorithm and demonstrate the potential advantages of combining dry-like friction with extrapolation and line search techniques.

math.OC↗
arXiv2025

Analysis of the Impact of the Union Budget Announcements on the Indian Stock Market: A Fractal Perspective

The stock market closely monitors macroeconomic policy announcements, such as annual budget events, due to their substantial influence on various economic participants. These events tend to impact the stock markets initially before affecting the real sector. Our study aims to analyze the effects of the budget on the Indian stock market, specifically focusing on the announcement for the year 2024. We will compare this with the years 2023, 2022, and 2020, assessing its impact on the NIFTY50 index using average abnormal return (AAR) and cumulative average abnormal return (CAAR) over a period of -15 and +15 days, including the budget day. This study utilizes an innovative approach involving the fractal interpolation function, paired with fractal dimensional analysis, to study the fluctuations arising from budget announcements. The fractal perspective on the data offers an effective framework for understanding complex variations.

q-fin.ST↗
arXiv2024

Adaptive Stabilization Based on Machine Learning for Column Generation

Column generation (CG) is a well-established method for solving large-scale linear programs. It involves iteratively optimizing a subproblem containing a subset of columns and using its dual solution to generate new columns with negative reduced costs. This process continues until the dual values converge to the optimal dual solution to the original problem. A natural phenomenon in CG is the heavy oscillation of the dual values during iterations, which can lead to a substantial slowdown in the convergence rate. Stabilization techniques are devised to accelerate the convergence of dual values by using information beyond the state of the current subproblem. However, there remains a significant gap in obtaining more accurate dual values at an earlier stage. To further narrow this gap, this paper introduces a novel approach consisting of 1) a machine learning approach for accurate prediction of optimal dual solutions and 2) an adaptive stabilization technique that effectively capitalizes on accurate predictions. On the graph coloring problem, we show that our method achieves a significantly improved convergence rate compared to traditional methods.

math.OC↗
arXiv2024

Bregman Proximal Linearized ADMM for Minimizing Separable Sums Coupled by a Difference of Functions

In this paper, we develop a splitting algorithm incorporating Bregman distances to solve a broad class of linearly constrained composite optimization problems, whose objective function is the separable sum of possibly nonconvex nonsmooth functions and a smooth function, coupled by a difference of functions. This structure encapsulates numerous significant nonconvex and nonsmooth optimization problems in the current literature including the linearly constrained difference-of-convex problems. Relying on the successive linearization and alternating direction method of multipliers (ADMM), the proposed algorithm exhibits the global subsequential convergence to a stationary point of the underlying problem. We also establish the convergence of the full sequence generated by our algorithm under the Kurdyka--Lojasiewicz property and some mild assumptions. The efficiency of the proposed algorithm is tested on a robust principal component analysis problem and a nonconvex optimal power flow problem.

math.OC↗
arXiv2023

Adaptive Solution Prediction for Combinatorial Optimization

This paper aims to predict optimal solutions for combinatorial optimization problems (COPs) via machine learning (ML). To find high-quality solutions efficiently, existing work uses a ML prediction of the optimal solution to guide heuristic search, where the ML model is trained offline under the supervision of solved problem instances with known optimal solutions. To predict the optimal solution with sufficient accuracy, it is critical to provide a ML model with adequate features that can effectively characterize decision variables. However, acquiring such features is challenging due to the high complexity of COPs. This paper proposes a framework that can better characterize decision variables by harnessing feedback from a heuristic search over several iterative steps, enabling an offline-trained ML model to predict the optimal solution in an adaptive manner. We refer to this approach as adaptive solution prediction (ASP). Specifically, we employ a set of statistical measures as features, which can extract useful information from feasible solutions found by a heuristic search and inform the ML model as to which value a decision variable is likely to take in high-quality solutions. Our experiments on three NP-hard COPs show that ASP substantially improves the prediction quality of an offline-trained ML model and achieves competitive results compared to several heuristic methods in terms of solution quality. Furthermore, we demonstrate that ASP can be used as a heuristic-pricing method for column generation, to boost an exact branch-and-price algorithm for solving the graph coloring problem.

math.OC↗
arXiv2022

Enhancing Column Generation by a Machine-Learning-Based Pricing Heuristic for Graph Coloring

Column Generation (CG) is an effective method for solving large-scale optimization problems. CG starts by solving a sub-problem with a subset of columns (i.e., variables) and gradually includes new columns that can improve the solution of the current subproblem. The new columns are generated as needed by repeatedly solving a pricing problem, which is often NP-hard and is a bottleneck of the CG approach. To tackle this, we propose a Machine-Learning-based Pricing Heuristic (MLPH)that can generate many high-quality columns efficiently. In each iteration of CG, our MLPH leverages an ML model to predict the optimal solution of the pricing problem, which is then used to guide a sampling method to efficiently generate multiple high-quality columns. Using the graph coloring problem, we empirically show that MLPH significantly enhancesCG as compared to six state-of-the-art methods, and the improvement in CG can lead to substantially better performance of the branch-and-price exact method.

math.OC↗
arXiv2021

Learning Primal Heuristics for Mixed Integer Programs

This paper proposes a novel primal heuristic for Mixed Integer Programs, by employing machine learning techniques. Mixed Integer Programming is a general technique for formulating combinatorial optimization problems. Inside a solver, primal heuristics play a critical role in finding good feasible solutions that enable one to tighten the duality gap from the outset of the Branch-and-Bound algorithm (B&B), greatly improving its performance by pruning the B&B tree aggressively. In this paper, we investigate whether effective primal heuristics can be automatically learned via machine learning. We propose a new method to represent an optimization problem as a graph, and train a Graph Convolutional Network on solved problem instances with known optimal solutions. This in turn can predict the values of decision variables in the optimal solution for an unseen problem instance of a similar type. The prediction of variable solutions is then leveraged by a novel configuration of the B&B method, Probabilistic Branching with guided Depth-first Search (PB-DFS) approach, aiming to find (near-)optimal solutions quickly. The experimental results show that this new heuristic can find better primal solutions at a much earlier stage of the solving process, compared to other state-of-the-art primal heuristics.

cs.AI↗
arXiv2021

Convergence Rates for Boundedly Regular Systems

In this work, we consider a continuous dynamical system associated with the fixed point set of a nonexpansive operator which was originally studied by Boţ & Csetnek (2015). Our main results establish convergence rates for the system's trajectories when the nonexpansive operator satisfies an additional regularity property. This setting is the natural continuous-time analogue to discrete-time results obtained in Bauschke, Noll & Phan (2015) and Borwein, Li & Tam (2017) by using the same regularity properties.

math.OC↗
arXiv2017

Outer limits of subdifferentials for min-max type functions

We generalise the outer subdifferential constructon suggested by Cánovas, Henrion, López and Parra for max type functions to pointwise minima of regular Lipschitz functions. We also answer an open question about the relation between the outer subdifferential of the support of a regular function and the end set of its subdifferential posed by Li, Meng and Yang.

math.OC↗
arXiv2017

Combining Progressive Hedging with a Frank-Wolfe Method to Compute Lagrangian Dual Bounds in Stochastic Mixed-Integer Programming

We present a new primal-dual algorithm for computing the value of the Lagrangian dual of a stochastic mixed-integer program (SMIP) formed by relaxing its nonanticipativity constraints. This dual is widely used in decomposition methods for the solution of SMIPs. The algorithm relies on the well-known progressive hedging method, but unlike previous progressive hedging approaches for SMIP, our algorithm can be shown to converge to the optimal Lagrangian dual value. The key improvement in the new algorithm is an inner loop of optimized linearization steps, similar to those taken in the classical Frank-Wolfe method. Numerical results demonstrate that our new algorithm empirically outperforms the standard implementation of progressive hedging for obtaining bounds in SMIP.

math.OC↗
arXiv2017

A parallelizable augmented Lagrangian method applied to large-scale non-convex-constrained optimization problems

We contribute improvements to a Lagrangian dual solution approach applied to large-scale optimization problems whose objective functions are convex, continuously differentiable and possibly nonlinear, while the non-relaxed constraint set is compact but not necessarily convex. Such problems arise, for example, in the split-variable deterministic reformulation of stochastic mixed-integer optimization problems. The dual solution approach needs to address the nonconvexity of the non-relaxed constraint set while being efficiently implementable in parallel. We adapt the augmented Lagrangian method framework to address the presence of nonconvexity in the non-relaxed constraint set and the need for efficient parallelization. The development of our approach is most naturally compared with the development of proximal bundle methods and especially with their use of serious step conditions. However, deviations from these developments allow for an improvement in efficiency with which parallelization can be utilized. Pivotal in our modification to the augmented Lagrangian method is the use of an integration of approaches based on the simplicial decomposition method (SDM) and the nonlinear block Gauss-Seidel (GS) method. An adaptation of a serious step condition associated with proximal bundle methods allows for the approximation tolerance to be automatically adjusted. Under mild conditions optimal dual convergence is proven, and we report computational results on test instances from the stochastic optimization literature. We demonstrate improvement in parallel speedup over a baseline parallel approach.

math.OC↗
arXiv2017

A Relax-and-Decomposition Algorithm for a p-Robust Hub Location Problem

In this paper, a non-linear p-robust hub location problem is extended to a risky environment where augmented chance constraint with a min-max regret form is employed to consider network risk as one of the objectives. The model considers risk factors such as security, air pollution and congestion to design the robust hub network. A Monte-Carlo simulation based algorithm, namely, a sample average approximation scheme is applied to select a set of efficient scenarios. The problem is then solved using a novel relax-and-decomposition heuristic based on the coupling of an accelerated Benders decomposition with a Lagrangian relaxation method. To improve the decomposition mechanism, a multi-Pareto cut version is applied in the proposed algorithm. In our numerical tests a modification of the well-known CAB data set is used with different levels of parameters uncertainty. The results demonstrate the capability of the proposed model to design a robust network. We also verify the accuracy of the sample average approximation method. Finally, the results of the proposed algorithm for different instances were compared to other solution approaches which confirm the efficiency of the proposed solution method.

math.OC↗
arXiv2017

Combining Penalty-based and Gauss-Seidel Methods for solving Stochastic Mixed-Integer Problems

In this paper, we propose a novel decomposition approach for mixed-integer stochastic programming (SMIP) problems that is inspired by the combination of penalty-based Lagrangian and block Gauss-Seidel methods (PBGS). In this sense, PBGS is developed such that the inherent decomposable structure that SMIPs present can be exploited in a computationally efficient manner. The performance of the proposed method is compared with the Progressive Hedging method (PH), which also can be viewed as a Lagrangian-based method for obtaining solutions for SMIP. Numerical experiments performed using instances from the literature illustrate the efficiency of the proposed method in terms of computational performance and solution quality.

math.OC↗
arXiv2015

The U-Lagrangian of a prox-regular function

When restricted to a subspace, a nonsmooth function can be differentiable. It is known that for a nonsmooth convex function f and a point x, the Euclidean space can be decomposed into two subspaces: U, over which a special Lagrangian can be defined and has nice smooth properties and V, the orthogonal complement subspace of U. In this paper we generalize the definition of UV-decomposition and U-Lagrangian to the context of nonconvex functions, specifically that of a prox-regular function.

math.OC↗
arXiv2015

Orbital Geometry in Optimisation

We discuss the use of group symmetries in optimisation, in particular with respect to the structure of subdifferential and projection operators. This allows us to generalise a classic result of Adrian Lewis regarding the characterisation of the subdifferential of a permutation invariant convex function to the characterisation of the proximal subdifferential of a Schur convex function that is invariant with respect to a finite reflection group. We are also able to simplify and generalise results on projections onto symmetric sets, in particular, we study projections on sparsity constraints used in sparse signal recovery and compressed sensing.

math.OC↗
arXiv2015

A version of bundle method with linear programming

Bundle methods have been intensively studied for solving both convex and nonconvex optimization problems. In most of the bundle methods developed thus far, at least one quadratic programming (QP) subproblem needs to be solved in each iteration. In this paper, we exploit the feasibility of developing a bundle algorithm that only solves linear subproblems. We start from minimization of a convex function and show that the sequence of major iterations converge to a minimizer. For nonconvex functions we consider functions that are locally Lipschitz continuous and prox-regular on a bounded level set, and minimize the cutting-plane model over a trust region with infinity norm. The para-convexity of such functions allows us to use the locally convexified model and its convexity properties. Under some conditions and assumptions, we study the convergence of the proposed algorithm through the outer semicontinuity of the proximal mapping. Encouraging results of preliminary numerical experiments on standard test sets are provided.

math.OC↗