Nonlinear programming

About: Nonlinear programming is a research topic. Over the lifetime, 19486 publications have been published within this topic receiving 656602 citations. The topic is also known as: non-linear programming & NLP.

A model algorithm for composite nondifferentiable optimization problems

Abstract: Composite functions ϕ(x)=f(x)+h(c(x)), where f and c are smooth and h is convex, encompass many nondifferentiable optimization problems of interest including exact penalty functions in nonlinear programming, nonlinear min-max problems, best nonlinear L 1, L 2 and L ∞ approximation and finding feasible points of nonlinear inequalities. The idea is used of making a linear approximation to c(x) whilst including second order terms in a quadratic approximation to f(x). This is used to determine a composite function ψ which approximates ϕ(x) and a basic algorithm is proposed in which ψ is minimized on each iteration. If the technique of a step restriction (or trust region) is incorporated into the algorithm, then it is shown that global convergence can be proved. It is also described briefly how the above approximations ensure that a second order rate of convergence is achieved by the basic algorithm.

Algorithms for nonlinear bilevel mathematical programs

Abstract: The bilevel programming problem (BLPP) is a model of a leader-follower game in which play is sequential and cooperation is not permitted. Some basic properties of the general model are developed, and a conjecture relevant to solution procedures is presented. Two algorithms are presented for solving various versions of the game when certain convexity conditions hold. One algorithm relies upon a hybrid branch-and-bound scheme and does not guarantee global optimality. Another is based on objective function cuts and, barring numerical stability problems with the optimizer, is guaranteed to converge to an epsilon -optimal solution. The performance of the two algorithms is examined using randomly generated test problems. The computational performance of the branch-and-bound algorithm is explored, and the cutting-plane algorithm is used to determine whether or not the branch-and-bound algorithm is uncovering global optima. >

An algorithm for nonlinear optimization using linear programming and equality constrained subproblems

Abstract: This paper describes an active-set algorithm for large-scale nonlinear programming based on the successive linear programming method proposed by Fletcher and Sainz de la Maza [10]. The step computation is performed in two stages. In the first stage a linear program is solved to estimate the active set at the solution. The linear program is obtained by making a linear approximation to the l1 penalty function inside a trust region. In the second stage, an equality constrained quadratic program (EQP) is solved involving only those constraints that are active at the solution of the linear program. The EQP incorporates a trust-region constraint and is solved (inexactly) by means of a projected conjugate gradient method. Numerical experiments are presented illustrating the performance of the algorithm on the CUTEr [1, 15] test set.

Foundations of optimization

Abstract: I: Convex Analysis.- 1: Linear Subspaces and Affine Manifolds.- 1.1 Linear Subspaces and Orthogonal Complements.- 1.2 Linear Independence and Dimensionality.- 1.3 Projection Theorem.- 1.4 Affine Manifolds.- 2: Convex Sets.- 2.1 Convex Cones, Convex Sets and Convex Hills.- 2.2 Caratheodory Type Theorems.- 2.3 Relative Interior and Related Properties of Convex Sets.- 2.4 Support and Separation Theorems.- 3: Convex Cones.- 3.1 Cones, Convex Cones and Polar Cones.- 3.2 Polyhedral Cones.- 3.3 Cones Generated by Sets.- 3.4 Cone of Tangents.- 3.5 Cone of Attainable Directions, Cone of Feasible Directions and Cone of Interior Directions.- 4: Convex Functions.- 4.1 Definitions and Preliminary Results.- 4.2 Continuity and Directional Differentiability of Convex Functions.- 4.3 Differentiable Convex Functions.- 4.4 Some Examples of Convex Functions.- 4.5 Generalization of Convex Functions.- II: Optimality Conditions and Duality.- 5: Stationary Point Optimality Conditions with Differentiability.- 5.1 Inequality Constrained Problems.- 5.2 Inequality and Equality Constrained Problems.- 5.3 Optimality Criteria of the Minimum Principle Type.- 6: Constraint Qualifications.- 6.1 Inequality Constrained Problems.- 6.2 Equality and Inequality Constrained Problems.- 6.3 Necessary and Sufficient Qualification.- 7: Convex Programming without Differentiability.- 7.1 Saddle Point Optimality Criteria.- 7.2 Stationary Point Optimality Conditions.- 8: Lagrangian Duality.- 8.1 Definitions and Preliminary Results.- 8.2 The Strong Duality Theorem.- 9: Conjugate Duality.- 9.1 Closure of a Function.- 9.2 Conjugate Functions.- 9.3 Main Duality Theorem.- 9.4 Nonlinear Programming via Conjugate Functions.- Selected References.

On the Implementation of an Algorithm for Large-Scale Equality Constrained Optimization

Abstract: This paper describes a software implementation of Byrd and Omojokun's trust region algorithm for solving nonlinear equality constrained optimization problems. The code is designed for the efficient solution of large problems and provides the user with a variety of linear algebra techniques for solving the subproblems occurring in the algorithm. Second derivative information can be used, but when it is not available, limited memory quasi-Newton approximations are made. The performance of the code is studied using a set of difficult test problems from the CUTE collection.