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We study the problem of decomposing a polynomial p into a sum of r squares by minimizing a quadratically penalized objective fp(u)=‖‖∑ri=1u2i−p‖‖2. This objective is nonconvex and is equivalent to the rank-r Burer-Monteiro factorization of a semidefinite program (SDP) encoding the sum of squares decomposition. We show that for all univariate polynomials p, if r≥2 then fp(u) has no spurious second-order critical points, showing that all local optima are also global optima. This is in contrast to previous work showing that for general SDPs, in addition to genericity conditions, r has to be roughly the square root of the number of constraints (the degree of p) for there to be no spurious second-order critical points. Our proof uses tools from computational algebraic geometry and can be interpreted as constructing a certificate using the first- and second-order necessary conditions. We also show that by choosing a norm based on sampling equally-spaced points on the circle, the gradient ∇fp can be computed in nearly linear time using fast Fourier transforms. Experimentally we demonstrate that this method has very fast convergence using first-order optimization algorithms such as L-BFGS, with near-linear scaling to million-degree polynomials. 

JuMP is an algebraic modeling language embedded in the Julia programming language. JuMP allows users to model optimization problems of a variety of kinds, including linear programming, integer programming, conic optimization, semidefinite programming, and nonlinear programming, and handles the low-level details of communicating with solvers. After nearly 10 years in development, JuMP 1.0 was released in March, 2022. In this short communication, we highlight the improvements to JuMP from recent releases up to and including 1.0. 

Spectral functions on Euclidean Jordan algebras arise frequently in convex optimization models. Despite the success of primal-dual conic interior point solvers, there has been little work on enabling direct support for spectral cones, that is, proper nonsymmetric cones defined from epigraphs and perspectives of spectral functions. We propose simple logarithmically homogeneous barriers for spectral cones and we derive efficient, numerically stable procedures for evaluating barrier oracles such as inverse Hessian operators. For two useful classes of spectral cones—the root-determinant cones and the matrix monotone derivative cones—we show that the barriers are self-concordant, with nearly optimal parameters. We implement these cones and oracles in our open-source solver Hypatia, and we write simple, natural formulations for four applied problems. Our computational benchmarks demonstrate that Hypatia often solves the natural formulations more efficiently than advanced solvers such as MOSEK 9 solve equivalent extended formulations written using only the cones these solvers support. Funding: This work was supported by Office of Naval Research [Grant N00014-18-1-2079] and the National Science Foundation [Grant OAC-1835443]. 

Automatic differentiation (AD), a technique for constructing new programs which compute the derivative of an original program, has become ubiquitous throughout scientific computing and deep learning due to the improved performance afforded by gradient-based optimization. However, AD systems have been restricted to the subset of programs that have a continuous dependence on parameters. Programs that have discrete stochastic behaviors governed by distribution parameters, such as flipping a coin with probability p of being heads, pose a challenge to these systems because the connection between the result (heads vs tails) and the parameters ( p ) is fundamentally discrete. In this paper we develop a new reparameterization-based methodology that allows for generating programs whose expectation is the derivative of the expectation of the original program. We showcase how this method gives an unbiased and low-variance estimator which is as automated as traditional AD mechanisms. We demonstrate unbiased forward-mode AD of discrete-time Markov chains, agent-based models such as Conway's Game of Life, and unbiased reverse-mode AD of a particle filter. Our code package is available at https://github.com/gaurav-arya/StochasticAD.jl. 

Abstract In recent work, we provide computational arguments for expanding the class of proper cones recognized by conic optimization solvers, to permit simpler, smaller, more natural conic formulations. We define an exotic cone as a proper cone for which we can implement a small set of tractable (i.e. fast, numerically stable, analytic) oracles for a logarithmically homogeneous self-concordant barrier for the cone or for its dual cone. Our extensible, open-source conic interior point solver, Hypatia, allows modeling and solving any conic problem over a Cartesian product of exotic cones. In this paper, we introduce Hypatia’s interior point algorithm, which generalizes that of Skajaa and Ye (Math. Program. 150(2):391–422, 2015) by handling exotic cones without tractable primal oracles. To improve iteration count and solve time in practice, we propose four enhancements to the interior point stepping procedure of Skajaa and Ye: (1) loosening the central path proximity conditions, (2) adjusting the directions using a third order directional derivative barrier oracle, (3) performing a backtracking search on a curve, and (4) combining the prediction and centering directions. We implement 23 useful exotic cones in Hypatia. We summarize the complexity of computing oracles for these cones and show that our new third order oracle is not a bottleneck. From 37 applied examples, we generate a diverse benchmark set of 379 problems. Our computational testing shows that each stepping enhancement improves Hypatia’s iteration count and solve time. Altogether, the enhancements reduce the geometric means of iteration count and solve time by over 80% and 70% respectively.