arXiv subjects

Chengfeng Shen

Publications and source records attributed to Chengfeng Shen.

5 recordsLinked to original sources

arXiv2025

Recursive contracts in non-convex environments

In this paper we examine non-convex dynamic optimization problems with forward looking constraints. We prove that the recursive multiplier formulation in \cite{marcet2019recursive} gives the optimal value if one assumes that the planner has access to a public randomization device and forward looking constraints only have to hold in expectations. Whether one formulates the functional equation as a sup-inf problem or as an inf-sup problem is essential for the timing of the optimal lottery and for determining which constraints have to hold in expectations. We discuss for which economic problems the use of lotteries can be considered a reasonable assumption. We provide a general method to recover the optimal policy from a solution of the functional equation. As an application of our results, we consider the Ramsey problem of optimal government policy and give examples where lotteries are essential for the optimal solution.

econ.TH↗

arXiv2025

A Lagrangian Approach to Optimal Randomization

We develop an efficient method for solving non-convex constrained optimization problems that are pervasive in economics. The optimal solution to these problems often involves randomization. We employ a Lagrangian framework and prove that the value of the saddle point characterizing the optimal random solution equals the value of the deterministic dual problem. Our algorithm solves this dual via subgradient descent and recovers the optimal random solution directly from deterministic optima computed along the iterations. For many non-convex economic problems, our method is orders of magnitude faster than linear programming, making previously intractable lottery problems feasible. As an application, we solve for optimal Mirrleesian income taxation with multi-dimensional types. We show that heterogeneity in productivity and Frisch elasticity can make randomization welfare-improving over the optimal deterministic schedule.

econ.TH↗

arXiv2023

Prompt Engineering Through the Lens of Optimal Control

Prompt Engineering (PE) has emerged as a critical technique for guiding Large Language Models (LLMs) in solving intricate tasks. Its importance is highlighted by its potential to significantly enhance the efficiency and effectiveness of human-machine interaction. As tasks grow increasingly complex, recent advanced PE methods have extended beyond the limitations of single-round interactions to embrace multi-round interactions, which allows for a deeper and more nuanced engagement with LLMs. In this paper, we propose an optimal control framework tailored for multi-round interactions with LLMs. This framework provides a unified mathematical structure that not only systematizes the existing PE methods but also sets the stage for rigorous analytical improvements. Furthermore, we extend this framework to include PE via ensemble methods and multi-agent collaboration, thereby enlarging the scope of applicability. By adopting an optimal control perspective, we offer fresh insights into existing PE methods and highlight theoretical challenges that warrant future research. Besides, our work lays a foundation for the development of more effective and interpretable PE methods.

cs.LG↗

arXiv2023

Fictitious Play for Mean Field Games with Optimal Stopping: Convergence and Computation

This paper studies mean field games with optimal stopping time (OSMFGs) where agents make optimal exit decisions. Such a model features a coupled obstacle problem and Fokker-Planck equation posing challenges on top of classic mean field games. The nonconvex nature of exit decisions renders the existence of a classic pure strategy equilibrium infeasible, necessitating the consideration of more complex mixed strategy equilibria. This paper proposes a generalized fictitious play algorithm that computes OSMFG mixed equilibria by iteratively solving pure strategy systems, i.e., approximating mixed strategies through averaging pure strategies according to a certain updating rule. The generalized fictitious play allows for a broad family of learning rates and the convergence to the mixed strategy equilibrium can be rigorously justified. The algorithm also incorporates efficient finite difference schemes of the pure strategy system. Numerical experiments demonstrate the effectiveness of the proposed method in robustly and efficiently computing mixed equilibria for OSMFGs.

math.NA↗

arXiv2023

Tumor growth with a necrotic core as an obstacle problem in pressure

Motivated by the incompressible limit of a cell density model, we propose a free boundary tumor growth model where the pressure satisfies an obstacle problem on an evolving domain $\Omega(t)$, and the coincidence set $\Lambda(t)$ captures the emerging necrotic core. We contribute to the analytical characterization of the solution structure in the following two aspects. By deriving a semi-analytical solution and studying its dynamical behavior, we obtain quantitative transitional properties of the solution separating phases in the development of necrotic cores and establish its long time limit with the traveling wave solutions. Also, we prove the existence of traveling wave solutions incorporating non-zero outer densities outside the tumor bulk, provided that the size of the outer density is below a threshold.

math.AP↗