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作者：Levin, Eitan; Kileel, Joe; Boumal, Nicolas
作者单位：California Institute of Technology; University of Texas System; University of Texas Austin; University of Texas System; University of Texas Austin; Swiss Federal Institutes of Technology Domain; Ecole Polytechnique Federale de Lausanne
摘要：We develop new tools to study landscapes in nonconvex optimization. Given one optimization problem, we pair it with another by smoothly parametrizing the domain. This is either for practical purposes (e.g., to use smooth optimization algorithms with good guarantees) or for theoretical purposes (e.g., to reveal that the landscape satisfies a strict saddle property). In both cases, the central question is: how do the landscapes of the two problems relate? More precisely: how do desirable points ...
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作者：Hollender, Alexandros; Zampetakis, Manolis
作者单位：University of Oxford; Yale University
摘要：Finding approximate stationary points, i.e., points where the gradient is approximately zero, of non-convex but smooth objective functions f over unrestricted d-dimensional domains is one of the most fundamental problems in classical non-convex optimization. Nevertheless, the computational and query complexity of this problem are still not well understood when the dimension d of the problem is independent of the approximation error. In this paper, we show the following computational and query ...
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作者：Xu, Luze; Lee, Jon
作者单位：University of California System; University of California Davis; University of Michigan System; University of Michigan
摘要：Mixed-integer nonlinear optimization formulations of the disjunction between the origin and a polytope via a binary indicator variable is broadly used in nonlinear combinatorial optimization for modeling a fixed cost associated with carrying out a group of activities and a convex cost function associated with the levels of the activities. The perspective relaxation of such models is often used to solve to global optimality in a branch-and-bound context, but it typically requires suitable conic...
