arXiv subjects

Negar Soheili

Publications and source records attributed to Negar Soheili.

9 recordsLinked to original sources

arXiv2026

Weakly Time-Coupled Approximation of Markov Decision Processes

Finite-horizon Markov decision processes (MDPs) with high-dimensional exogenous uncertainty and endogenous states arise in operations and finance, including the valuation and exercise of Bermudan and real options, but face a scalability barrier as computational complexity grows with the horizon. A common approximation represents the value function using basis functions, but methods for fitting weights treat cross-stage optimization differently. Least squares Monte Carlo (LSM) fits weights via backward recursion and regression, avoiding joint optimization but accumulating error over the horizon. Approximate linear programming (ALP) and pathwise optimization (PO) jointly fit weights to produce upper bounds, but temporal coupling causes computational complexity to grow with the horizon. We show this coupling is an artifact of the approximation architecture, and develop a weakly time-coupled approximation (WTCA) where cross-stage dependence is independent of horizon. For any fixed basis function set, the WTCA upper bound is tighter than that of ALP and looser than that of PO, and converges to the optimal policy value as the basis family expands. We extend parallel deterministic block coordinate descent to the stochastic MDP setting exploiting weak temporal coupling. Applied to WTCA, weak coupling yields computational complexity independent of the horizon. Within equal time budget, solving WTCA accommodates more exogenous samples or basis functions than PO, yielding tighter bounds despite PO being tighter for fixed samples and basis functions. On Bermudan option and ethanol production instances, WTCA produces tighter upper bounds than PO and LSM in every instance tested, with near-optimal policies at longer horizons.

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arXiv2021

Projection and rescaling algorithm for finding maximum support solutions to polyhedral conic systems

We propose a simple projection and rescaling algorithm that finds maximum support solutions to the pair of feasibility problems \[ \text{find} \; x\in L\cap\mathbb{R}^n_{+} \;\;\;\; \text{ and } \; \;\;\;\; \text{find} \; \hat x\in L^\perp\cap\mathbb{R}^n_{+}, \] where $L$ is a linear subspace of $\mathbb{R}^n$ and $L^\perp$ is its orthogonal complement. The algorithm complements a basic procedure that involves only projections onto $L$ and $L^\perp$ with a periodic rescaling step. The number of rescaling steps and thus overall computational work performed by the algorithm are bounded above in terms of a condition measure of the above pair of problems. Our algorithm is a natural but significant extension of a previous projection and rescaling algorithm that finds a solution to the problem \[ \text{find} \; x\in L\cap\mathbb{R}^n_{++} \] when this problem is feasible. As a byproduct of our new developments, we obtain a sharper analysis of the projection and rescaling algorithm in the latter special case.

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arXiv2021

Self-guided Approximate Linear Programs

Approximate linear programs (ALPs) are well-known models based on value function approximations (VFAs) to obtain policies and lower bounds on the optimal policy cost of discounted-cost Markov decision processes (MDPs). Formulating an ALP requires (i) basis functions, the linear combination of which defines the VFA, and (ii) a state-relevance distribution, which determines the relative importance of different states in the ALP objective for the purpose of minimizing VFA error. Both these choices are typically heuristic: basis function selection relies on domain knowledge while the state-relevance distribution is specified using the frequency of states visited by a heuristic policy. We propose a self-guided sequence of ALPs that embeds random basis functions obtained via inexpensive sampling and uses the known VFA from the previous iteration to guide VFA computation in the current iteration. Self-guided ALPs mitigate the need for domain knowledge during basis function selection as well as the impact of the initial choice of the state-relevance distribution, thus significantly reducing the ALP implementation burden. We establish high probability error bounds on the VFAs from this sequence and show that a worst-case measure of policy performance is improved. We find that these favorable implementation and theoretical properties translate to encouraging numerical results on perishable inventory control and options pricing applications, where self-guided ALP policies improve upon policies from problem-specific methods. More broadly, our research takes a meaningful step toward application-agnostic policies and bounds for MDPs.

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arXiv2020

An Adaptive Parameter-free and Projection-free Restarting Level Set Method for Constrained Convex Optimization Under the Error Bound Condition

Recent efforts to accelerate first-order methods have focused on convex optimization problems that satisfy a geometric property known as error-bound condition, which covers a broad class of problems, including piece-wise linear programs and strongly convex programs. Parameter-free first-order methods that employ projection-free updates have the potential to broaden the benefit of acceleration. Such a method has been developed for unconstrained convex optimization but is lacking for general constrained convex optimization. We propose a parameter-free level-set method for the latter constrained case based on projection-free subgradient method that exhibits accelerated convergence for problems that satisfy an error-bound condition. Our method maintains a separate copy of the level-set sub-problem for each level parameter value and restarts the computation of these copies based on objective function progress. Applying such a restarting scheme in a level-set context is novel and results in an algorithm that dynamically adapts the precision of each copy. This property is key to extending prior restarting methods based on static precision that have been proposed for unconstrained convex optimization to handle constraints. We report promising numerical performance relative to benchmark methods.

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arXiv2020

A Data Efficient and Feasible Level Set Method for Stochastic Convex Optimization with Expectation Constraints

Stochastic convex optimization problems with expectation constraints (SOECs) are encountered in statistics and machine learning, business, and engineering. In data-rich environments, the SOEC objective and constraints contain expectations defined with respect to large datasets. Therefore, efficient algorithms for solving such SOECs need to limit the fraction of data points that they use, which we refer to as algorithmic data complexity. Recent stochastic first order methods exhibit low data complexity when handling SOECs but guarantee near-feasibility and near-optimality only at convergence. These methods may thus return highly infeasible solutions when heuristically terminated, as is often the case, due to theoretical convergence criteria being highly conservative. This issue limits the use of first order methods in several applications where the SOEC constraints encode implementation requirements. We design a stochastic feasible level set method (SFLS) for SOECs that has low data complexity and emphasizes feasibility before convergence. Specifically, our level-set method solves a root-finding problem by calling a novel first order oracle that computes a stochastic upper bound on the level-set function by extending mirror descent and online validation techniques. We establish that SFLS maintains a high-probability feasible solution at each root-finding iteration and exhibits favorable iteration complexity compared to state-of-the-art deterministic feasible level set and stochastic subgradient methods. Numerical experiments on three diverse applications validate the low data complexity of SFLS relative to the former approach and highlight how SFLS finds feasible solutions with small optimality gaps significantly faster than the latter method.

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arXiv2019

Computational performance of a projection and rescaling algorithm

This paper documents a computational implementation of a {\em projection and rescaling algorithm} for finding most interior solutions to the pair of feasibility problems \[ \text{find} \; x\in L\cap\mathbb{R}^n_{+} \;\;\;\; \text{ and } \; \;\;\;\; \text{find} \; \hat x\in L^\perp\cap\mathbb{R}^n_{+}, \] where $L$ denotes a linear subspace in $\mathbb{R}^n$ and $L^\perp$ denotes its orthogonal complement. The projection and rescaling algorithm is a recently developed method that combines a {\em basic procedure} involving only low-cost operations with a periodic {\em rescaling step.} We give a full description of a MATLAB implementation of this algorithm and present multiple sets of numerical experiments on synthetic problem instances with varied levels of conditioning. Our computational experiments provide promising evidence of the effectiveness of the projection and rescaling algorithm. Our MATLAB code is publicly available. Furthermore, the simplicity of the algorithm makes a computational implementation in other environments completely straightforward.

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arXiv2016

Solving Conic Systems via Projection and Rescaling

We propose a simple projection and rescaling algorithm to solve the feasibility problem \[ \text{ find } x \in L \cap Ω, \] where $L$ and $Ω$ are respectively a linear subspace and the interior of a symmetric cone in a finite-dimensional vector space $V$. This projection and rescaling algorithm is inspired by previous work on rescaled versions of the perceptron algorithm and by Chubanov's projection-based method for linear feasibility problems. As in these predecessors, each main iteration of our algorithm contains two steps: a {\em basic procedure} and a {\em rescaling} step. When $L \cap Ω\ne \emptyset$, the projection and rescaling algorithm finds a point $x \in L \cap Ω$ in at most $O(\log(1/δ(L \cap Ω)))$ iterations, where $δ(L \cap Ω) \in (0,1]$ is a measure of the most interior point in $L \cap Ω$. The ideal value $δ(L\cap Ω) = 1$ is attained when $L \cap Ω$ contains the center of the symmetric cone $Ω$. We describe several possible implementations for the basic procedure including a perceptron scheme and a smooth perceptron scheme. The perceptron scheme requires $O(r^4)$ perceptron updates and the smooth perceptron scheme requires $O(r^2)$ smooth perceptron updates, where $r$ stands for the Jordan algebra rank of $V$.

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arXiv2015

On the von Neumann and Frank-Wolfe Algorithms with Away Steps

The von Neumann algorithm is a simple coordinate-descent algorithm to determine whether the origin belongs to a polytope generated by a finite set of points. When the origin is in the of the polytope, the algorithm generates a sequence of points in the polytope that converges linearly to zero. The algorithm's rate of convergence depends on the radius of the largest ball around the origin contained in the polytope. We show that under the weaker condition that the origin is in the polytope, possibly on its boundary, a variant of the von Neumann algorithm that includes generates a sequence of points in the polytope that converges linearly to zero. The new algorithm's rate of convergence depends on a certain geometric parameter of the polytope that extends the above radius but is always positive. Our linear convergence result and geometric insights also extend to a variant of the Frank-Wolfe algorithm with away steps for minimizing a strongly convex function over a polytope.

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arXiv2013

Some preconditioners for systems of linear inequalities

We show that a combination of two simple preprocessing steps would generally improve the conditioning of a homogeneous system of linear inequalities. Our approach is based on a comparison among three different but related notions of conditioning for linear inequalities.

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