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1. Level constrained first order methods for function constrained optimization

   https://doi.org/10.1007/s10107-024-02057-4  

   Boob, Digvijay; Deng, Qi; Lan, Guanghui (March 2024, Mathematical Programming)

   Abstract We present a new feasible proximal gradient method for constrained optimization where both the objective and constraint functions are given by summation of a smooth, possibly nonconvex function and a convex simple function. The algorithm converts the original problem into a sequence of convex subproblems. Formulating those subproblems requires the evaluation of at most one gradient-value of the original objective and constraint functions. Either exact or approximate subproblems solutions can be computed efficiently in many cases. An important feature of the algorithm is the constraint level parameter. By carefully increasing this level for each subproblem, we provide a simple solution to overcome the challenge of bounding the Lagrangian multipliers and show that the algorithm follows a strictly feasible solution path till convergence to the stationary point. We develop a simple, proximal gradient descent type analysis, showing that the complexity bound of this new algorithm is comparable to gradient descent for the unconstrained setting which is new in the literature. Exploiting this new design and analysis technique, we extend our algorithms to some more challenging constrained optimization problems where (1) the objective is a stochastic or finite-sum function, and (2) structured nonsmooth functions replace smooth components of both objective and constraint functions. Complexity results for these problems also seem to be new in the literature. Finally, our method can also be applied to convex function constrained problems where we show complexities similar to the proximal gradient method. 

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2. Constrained Information Design

   https://doi.org/10.1287/moor.2022.1346  

   Doval, Laura; Skreta, Vasiliki (January 2023, Mathematics of Operations Research)

   We provide tools to analyze information design problems subject to constraints. We do so by extending an insight by Le Treust and Tomala to the case of multiple inequality and equality constraints. Namely, that an information design problem subject to constraints can be represented as an unconstrained information design problem with additional states, one for each constraint. Thus, without loss of generality, optimal solutions induce as many posteriors as the number of states and constraints. We provide results that refine this upper bound. Furthermore, we provide conditions under which there is no duality gap in constrained information design, thus validating a Lagrangian approach. We illustrate our results with applications to mechanism design with limited commitment and persuasion of a privately informed receiver. Funding: L. Doval acknowledges the support of the National Science Foundation through [Grant SES-2131706]. V. Skreta acknowledges the support from the National Science Foundation through [Grant SES-1851729] and from the European Research Council (ERC) through consolidator [Grant 682417]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/moor.2022.1346 . 

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3. Resilient Constrained Learning

   Hounie, Ignacio; Ribeiro, Alejandro; Chamon, Luiz F_O (December 2023, Advances in Neural Information Processing Systems)
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   Prabhanjan Ananth, Abhishek Jain (January 2022, 13th Innovations in Theoretical Computer Science Conference)
5. Constrained Retrospective Search

   https://doi.org/10.1257/pandp.20211054  

   Urgun, Can; Yariv, Leeat (May 2021, AEA Papers and Proceedings)

   null (Ed.)

   The search for good outcomes–be it government policies, technological breakthroughs, or lasting purchases–takes time and effort. At times, the decision process is unconstrained: an individual seeking a well-priced product determines her search scope and time as she wishes. At times, search is constrained, either through institutions or cognitive limitations. We consider retrospective search in both settings: an agent chooses the search scope and time, selecting the best observed outcome upon stopping. We analyze the impacts of constraints when observed samples are independent and correlated over time. 

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