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Saunders: Large-scale linearly constrained optimization. Programming 14 , 41 - Saunders: A projected Lagrangian algorithm and its implementation for sparse nonlinear constraints. Programming Study 16 , SOL R, Dept. Osterby, Z. Zlatev: Direct methods for sparse matrices. Pissanetzky: Sparse Matrix Technology.

Introduction to Numerical Optimization Gradient Descent - 1

Reid: On the method of conjugate gradients for the solution of large sparse systems of linear equations. Reid, ed. Harwell Reid: A sparsity-exploiting variant of the Bartels-Golub decomposition for linear programming bases. Programming 24 , Ritch: Discrete optimal control with multiple constraints I: Constraint separation and transformation technique. Automatica 9 , Zlatev: A survey of the advances in the exploitation of the sparsity in the solution of large problems. Congress on Comp.

Each component shows whether a corresponding constraint is active that is, whether a variable is at the bound :. Number of function evaluations done. Number of Jacobian evaluations done. It runs the Levenberg-Marquardt algorithm formulated as a trust-region type algorithm. The implementation is based on paper [JJMore] , it is very robust and efficient with a lot of smart tricks. It should be your first choice for unconstrained problems. The algorithm iteratively solves trust-region subproblems augmented by a special diagonal quadratic term and with trust-region shape determined by the distance from the bounds and the direction of the gradient.

This enhancements help to avoid making steps directly into bounds and efficiently explore the whole space of variables. To further improve convergence, the algorithm considers search directions reflected from the bounds. To obey theoretical requirements, the algorithm keeps iterates strictly feasible.

MS&E Large-Scale Numerical Optimization

The difference from the MINPACK implementation is that a singular value decomposition of a Jacobian matrix is done once per iteration, instead of a QR decomposition and series of Givens rotation eliminations. The subspace is spanned by a scaled gradient and an approximate Gauss-Newton solution delivered by scipy. The algorithm works quite robust in unbounded and bounded problems, thus it is chosen as a default algorithm.

The required Gauss-Newton step can be computed exactly for dense Jacobians or approximately by scipy. The algorithm is likely to exhibit slow convergence when the rank of Jacobian is less than the number of variables. Robust loss functions are implemented as described in [BA]. The idea is to modify a residual vector and a Jacobian matrix on each iteration such that computed gradient and Gauss-Newton Hessian approximation match the true gradient and Hessian approximation of the cost function.

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Then the algorithm proceeds in a normal way, i. Branch, T. Coleman, and Y. William H. Press et.

The Art of Scientific Computing. Byrd, R. Schnabel and G. Programming, 40, pp. Curtis, M.

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