Introduction to the Theory of Interior Point Methods

1996 ◽  
pp. 3-34
Author(s):  
Benjamin Jansen ◽  
Cornelis Roos ◽  
Tamás Terlaky
Keyword(s):  
Interior Point ◽  
Interior Point Methods

scholarly journals Approximate solution of system of equations arising in interior-point methods for bound-constrained optimization

2021 ◽  
Author(s):  
David Ek ◽  
Anders Forsgren
Keyword(s):  
Approximate Solution ◽  
Interior Point ◽  
Interior Point Methods ◽  
Linear Equations ◽  
Approximate Solutions ◽  
System Of Nonlinear Equations ◽  
Intermediate Step ◽  
System Of Linear Equations ◽  
Constrained Nonlinear Optimization ◽  
Asymptotic Error

AbstractThe focus in this paper is interior-point methods for bound-constrained nonlinear optimization, where the system of nonlinear equations that arise are solved with Newton’s method. There is a trade-off between solving Newton systems directly, which give high quality solutions, and solving many approximate Newton systems which are computationally less expensive but give lower quality solutions. We propose partial and full approximate solutions to the Newton systems. The specific approximate solution depends on estimates of the active and inactive constraints at the solution. These sets are at each iteration estimated by basic heuristics. The partial approximate solutions are computationally inexpensive, whereas a system of linear equations needs to be solved for the full approximate solution. The size of the system is determined by the estimate of the inactive constraints at the solution. In addition, we motivate and suggest two Newton-like approaches which are based on an intermediate step that consists of the partial approximate solutions. The theoretical setting is introduced and asymptotic error bounds are given. We also give numerical results to investigate the performance of the approximate solutions within and beyond the theoretical framework.

scholarly journals Contractors and Linear Matrix Inequalities

2015 ◽  
Vol 1 (3) ◽  
Author(s):  
Jeremy Nicola ◽  
Luc Jaulin
Keyword(s):  
Fixed Point ◽  
Large Class ◽  
Linear Matrix Inequalities ◽  
Interior Point ◽  
Interior Point Methods ◽  
Linear Constraints ◽  
Linear Matrix ◽  
Nonlinear Constraints ◽  
Matrix Inequalities ◽  
Convex Constraints

Linear matrix inequalities (LMIs) comprise a large class of convex constraints. Boxes, ellipsoids, and linear constraints can be represented by LMIs. The intersection of LMIs are also classified as LMIs. Interior-point methods are able to minimize or maximize any linear criterion of LMIs with complexity, which is polynomial regarding to the number of variables. As a consequence, as shown in this paper, it is possible to build optimal contractors for sets represented by LMIs. When solving a set of nonlinear constraints, one may extract from all constraints that are LMIs in order to build a single optimal LMI contractor. A combination of all contractors obtained for other non-LMI constraints can thus be performed up to the fixed point. The resulting propogation is shown to be more efficient than other conventional contractor-based approaches.

scholarly journals Hyperbolic Polynomials and Convex Analysis

2001 ◽  
Vol 53 (3) ◽  
pp. 470-488 ◽  
Author(s):  
Heinz H. Bauschke ◽  
Osman Güler ◽  
Adrian S. Lewis ◽  
Hristo S. Sendov
Keyword(s):  
Convex Function ◽  
Interior Point ◽  
Convex Analysis ◽  
Symmetric Functions ◽  
Interior Point Methods ◽  
Optimization Problems ◽  
Real Polynomial ◽  
Hyperbolic Polynomials ◽  
Real Roots ◽  
Univariate Polynomial

AbstractA homogeneous real polynomial p is hyperbolic with respect to a given vector d if the univariate polynomial t ⟼ p(x − td) has all real roots for all vectors x. Motivated by partial differential equations, Gårding proved in 1951 that the largest such root is a convex function of x, and showed various ways of constructing new hyperbolic polynomials. We present a powerful new such construction, and use it to generalize Gårding’s result to arbitrary symmetric functions of the roots. Many classical and recent inequalities follow easily. We develop various convex-analytic tools for such symmetric functions, of interest in interior-point methods for optimization problems over related cones.

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