MINOS. A Large-Scale Nonlinear Programming System (For Problems With Linear Constraints). User's Guide.

Abstract In this paper we consider large-scale smooth optimization problems with multiple linear coupled constraints. Due to the non-separability of the constraints, arbitrary random sketching would not be guaranteed to work. Thus, we first investigate necessary and sufficient conditions for the sketch sampling to have well-defined algorithms. Based on these sampling conditions we develop new sketch descent methods for solving general smooth linearly constrained problems, in particular, random sketch descent (RSD) and accelerated random sketch descent (A-RSD) methods. To our knowledge, this is the first convergence analysis of RSD algorithms for optimization problems with multiple non-separable linear constraints. For the general case, when the objective function is smooth and non-convex, we prove for the non-accelerated variant sublinear rate in expectation for an appropriate optimality measure. In the smooth convex case, we derive for both algorithms, non-accelerated and A-RSD, sublinear convergence rates in the expected values of the objective function. Additionally, if the objective function satisfies a strong convexity type condition, both algorithms converge linearly in expectation. In special cases, where complexity bounds are known for some particular sketching algorithms, such as coordinate descent methods for optimization problems with a single linear coupled constraint, our theory recovers the best known bounds. Finally, we present several numerical examples to illustrate the performances of our new algorithms.

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Large scale nonlinear programming

Computers & Chemical Engineering ◽

10.1016/0098-1354(83)80005-2 ◽

1983 ◽

Vol 7 (5) ◽

pp. 595-604 ◽

Cited By ~ 18

Author(s):

Leon S. Lasdon ◽

A.D. Waren

Keyword(s):

Nonlinear Programming ◽

Large Scale

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The Flattened Aggregate Constraint Homotopy Method for Nonlinear Programming Problems with Many Nonlinear Constraints

Abstract and Applied Analysis ◽

10.1155/2014/430932 ◽

2014 ◽

Vol 2014 ◽

pp. 1-14

Author(s):

Zhengyong Zhou ◽

Bo Yu

Keyword(s):

Nonlinear Programming ◽

Interior Point Method ◽

Large Scale ◽

State Of The Art ◽

Numerical Procedure ◽

Homotopy Method ◽

Nonlinear Constraints ◽

Computational Results ◽

Homotopy Methods ◽

Constraint Function

The aggregate constraint homotopy method uses a single smoothing constraint instead ofm-constraints to reduce the dimension of its homotopy map, and hence it is expected to be more efficient than the combined homotopy interior point method when the number of constraints is very large. However, the gradient and Hessian of the aggregate constraint function are complicated combinations of gradients and Hessians of all constraint functions, and hence they are expensive to calculate when the number of constraint functions is very large. In order to improve the performance of the aggregate constraint homotopy method for solving nonlinear programming problems, with few variables and many nonlinear constraints, a flattened aggregate constraint homotopy method, that can save much computation of gradients and Hessians of constraint functions, is presented. Under some similar conditions for other homotopy methods, existence and convergence of a smooth homotopy path are proven. A numerical procedure is given to implement the proposed homotopy method, preliminary computational results show its performance, and it is also competitive with the state-of-the-art solver KNITRO for solving large-scale nonlinear optimization.

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