We consider mixed-integer convex optimization problems in which binary indicators control continuous variables. We introduce the \emph{Coordinate Optimality Reformulation} (CORe) framework, which augments standard indicator formulations by incorporating coordinate-wise optimality information. The resulting reformulations preserve global optimality while substantially improving branch-and-bound performance, particularly in sparse and structured settings where the coordinate-wise optimality conditions expose exploitable problem structure. We first develop the main components of CORe, including coordinate-wise optimality conditions, closed-form characterizations, and disjunctive reformulations. We then demonstrate the framework across multiple problem families, including quadratic problems and robust single-index models. Computational experiments show that CORe can substantially improve solver performance compared with standard big-$M$ formulations.
Tong Xu, S. Fattahi, Andrés Gómez et al.· 0 citations
A unified framework, based on decision diagrams, is proposed that serves both to solve the associated optimization problems and to construct ideal conic quadratic extended formulations of the closure of the convex hull of the underlying mixed-integer set.
Soobin Choi, S. Fattahi, Andrés Gómez et al.· 1 citation
This paper reduces the inner worst-case expectation problem exactly to a scalar budget allocation task, and embeds this procedure within an oracle-based distributional best-response framework to directly compute an approximate primal-dual solution to the overall DRO problem.
Guixian Chen, S. Fattahi, Soroosh Shafiee· 1 citation
In nonsmooth optimization, identifiable sets describe the local region eventually reached by sequences converging to a prescribed critical point. When such a set is a $C^2$ manifold on which the objective restricts to a $C^2$ function, it is called an identifiable manifold. Their appeal lies in what they enable: many first-order methods identify these manifolds in finitely many iterations, after which the iterates enter a region in which the problem is effectively smooth. Consequently, many powerful tools and guarantees from smooth optimization transplant naturally to the nonsmooth setting. Owing to these properties, much existing work has focused on characterizing conditions that guarantee their existence. In this work, we study a complementary question: under what conditions is a critical point devoid of any identifiable manifold? We answer this by developing a deterministic branching criterion for a broad class of finite-max composite optimization problems, characterizing when a critical point admits no identifiable manifold. This criterion is surprisingly mild in certain classes of problems: it holds with high probability for overparameterized robust low-rank recovery and almost surely at common interpolators of random minimax regression, suggesting that the absence of identifiable manifolds may be the rule rather than the exception in modern optimization.
Yifan Wang, Jianhao Ma, S. Fattahi· 0 citations
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