# Ralph Baker KearfottUniversity of Louisiana at Lafayette | ULL · Department of Mathematics

Ralph Baker Kearfott

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186

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Introduction

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August 1977 - present

## Publications

Publications (186)

Mathematically rigorous global optimization and fuzzy optimization have different philosophical underpinnings, goals, and applications. However, some of the tools used in implementations are similar or identical. We review, compare and contrast basic ideas and applications behind these areas, referring to some of the work in the very large literatu...

We are concerned with tools to find bounds on the range of certain polynomial functions of n variables. Although our motivation and history of the tools are from crisp global optimization, bounding the range of such functions is also important in fuzzy logic implementations. We review and provide a new perspective on one such tool. We have been exa...

Linear program solvers sometimes fail to find a good approximation to the optimum value, without indicating possible failure. However, it may be important to know how close the value such solvers return is to an actual optimum, or even to obtain mathematically rigorous bounds on the optimum. In a seminal 2004 paper, Neumaier and Shcherbina, propose...

We discuss problems we have encountered when implementing algorithms and formulas for computing mathematically rigorous bounds on the optima of linear programs. The rigorous bounds are based on a 2004 idea of Neumaier and Shcherbina. In a note in the Journal of Global Optimization , we pointed out two minor, correctable but consequential and hard t...

In branch and bound algorithms in constrained global optimization, a sharp upper bound on the global optimum is important for the overall efficiency of the branch and bound process. Software to find local optimizers, using floating point arithmetic, often computes an approximately feasible point close to an actual global optimizer. Not mathematical...

In branch and bound algorithms for constrained global optimization, an acceleration technique is to construct regions \({\varvec{x}}^{*}\) around local optimizing points \(\check{x}\), then delete these regions from further search. The result of the algorithm is then a list of those small regions in which all globally optimizing points must lie. If...

The optimum and at least one optimizing point for convex nonlinear programs can be approximated well by the solution to a linear program a fact long used in branch and bound algorithms. In more general problems, we can identify subspaces of ‘non-convex variables’ such that, if these variables have sufficiently small ranges, the optimum and at least...

Minimax problems can be approached by reformulating them into smooth problems with constraints or by dealing with the non-smooth objective directly. We focus on verified enclosures of all globally optimal points of such problems. In smooth problems in branch and bound algorithms, interval Newton methods can be used to verify existence and uniquenes...

In previous work, we, and also Epperly and Pistikopoulos, proposed an analysis of general nonlinear programs that identified certain variables as convex, not ever needing subdivision, and non-convex, or possibly needing subdivision in branch and bound algorithms. We proposed a specific algorithm, based on a generated computational graph of the prob...

We provide a summary of the goals, underlying philosophy, work, decisions, product, and completion schedule of the IEEE P-1788 working group on interval arithmetic.

Mass traffic evacuations during Hurricanes Rita and Katrina demonstrated limitations of static planning-based evacuation models based on data from historical events. Evacuation dynamics are complex due to the number of people and vehicles, road networks, the uncertainty and perception of the event, public safety advisories, and human decisions rega...

We consider the general nonlinear program. Both linear and nonlinear programs are often approximately ill-posed, with an entire continuum of approximate optimizing points. As an example, we take a linear program derived from a simple investment scenario. In this problem, any point along the portion of a line is a solution to this problem. If we per...

Deterministic branch and bound methods for the solution of general nonlinear programs have become increasingly popular during the last decade or two, with increasing computer speed, algorithmic improvements, and multiprocessors. Presently, there are several commercial packages. Although such packages are based on exhaustive search, not all of them...

This is a correction to R. B. Kearfott and S. Hongthong's article [SIAM J. Optim., 16 (2005), pp. 418-433].

A standard for the notation of the most used quantities and operators in interval analysis is proposed.

We explain the installation and use of the GlobSol package for mathematically rigorous bounds on all solutions to constrained and unconstrained global optimization problems, as well as non-linear systems of equations. This document should be of use both to people with optimization problems to solve and to people incorporating GlobSol's components i...

A new Alternating-Direction Sinc-Galerkin (ADSG) method is developed and contrasted with classical Sinc-Galerkin methods. It is derived from an iterative scheme for solving the Lyapunov equation that arises when a symmetric Sinc-Galerkin method is used ...

The main goal of this introduction is to make the book more accessible to readerswho are not familiar with interval computations: to beginning graduate students, toresearchers from related fields, etc. With this goal in mind, this introduction describesthe basic ideas behind interval computations and behind the applications of intervalcomputations...

This volume deals, generally, with innovative techniques for automated knowledge representation and manipulation when such knowledge is subject to significant uncertainty, as well as with automated decision processes associated with such uncertain knowledge. Going beyond traditional probability theory and traditional statistical arguments, the tech...

This book is intended primarily for those not yet familiar with methods for computing with intervals of real numbers and what can be done with these methods.
Using a pair [a, b] of computer numbers to represent an interval of real numbers a ≤ x ≤ b, we define an arithmetic for intervals and interval valued extensions of functions commonly used in c...

Finding bounding sets to solutions to systems of algebraic equations with uncertainties in the coefficients, as well as rapidly but rigorously lo- cating all solutions to nonlinear systems or global optimization problems, involves bounding the solution sets to systems of equations with wide in- terval coefficients. In many cases, singular systems a...

Efforts have been made to standardize interval arithmetic (IA) for over a decade. The reasons have been to enable more widespread
use of the technology, to enable more widespread sharing and collaboration among researchers and developers of the technology,
and to enable easier checking that computer codes have been correctly programmed. During the...

The IFIP Working Group 2.5 on Numerical Software (IFIPWG2.5) wrote on 5th Septem- ber 2007 to the IEEE Standards Committee concerned with revising the IEEE Floating- Point Arithmetic Standards 754 and 854 (IEEE754R), expressing the unanimous request of IFIPWG2.5 that the following requirement be included in the future computer arithmetic standard:...

Keywords
The Basic Branch and Bound Algorithm for Unconstrained Optimization
Acceleration Tools
Differences Between Unconstrained and Constrained Optimization
See also
References

Keywords
Classical Fixed Point Theory and Interval Arithmetic
The Krawczyk Method and Fixed Point Theory
Interval Newton Methods and Fixed Point Theory
Uniqueness
Infinite-Dimensional Problems
See also
References

Keywords
Use In Automatic Differentiation
Use In Constraint Satisfaction Techniques.
Use In Symbolic Preprocessing.
See also
References

This article outlines how thiscan be done.For the fundamental concepts used throughoutthis article, see Interval analysis : Introduction,interval numbers and basic propertiesof interval arithmetic.General feasibility: the Fritz-John conditions.

Introduction.Nondifferentiable problems arise in variousplaces in global optimization. One example isin l 1 and l 1 optimization. That is,minxOE(x) = min kFk 1 = minxmXi=1jf i (x)j (1)andminxOE(x) = minkFk 1 = minx\Phimax1imjf i (x)j\Psi(2)where x is an n-vector, arise in data fitting,etc., and OE has a discontinuous gradient. Inother problems, pie...

Keywords
Introduction
Univariate Interval Newton Methods
Multivariate Interval Newton Methods
Existence-Proving Properties
See also
References

1~/($)(x), where ~(x) = ~ FT(x)F(x), may be numerically singular, so that it is appropriate to think of the least-squares solution to F(x) = 0 m as a curve in R n. These cases correspond to the previously mentioned artificial homotopy and naturally occurring parametrized system, respectively. The usual methods for solving non-parametrized nonlinear...

Structural engineers use design codes formulated to consider uncertainty for both reinforced concrete and structural steel design. For a simple one-bay structural steel frame, we survey typical un-certainties and compute an interval solution for displacements and forces. The naive solutions have large over-estimations, so we explore the Mullen-Muha...

Algorithms and comparison results for a derivative-free predictor-corrector method for following arcs of H(x,t) = ϑ, where
H : Rn × [0, 1] → Rn is smooth, are given. The method uses a least-change secant update for H', adaptive controlled predictor stepsize, and Powell's
indexing procedure to preserve linear independence in the updates. Considerabl...

Both theory and implementations in deterministic global optimization have advanced significantly in the past decade. Two schools of thought have developed: the first employs various bounding techniques without validation, while the second employs different techniques, in a way that always rigorously takes account of roundoff error (i.e. with valida...

We propose the collection, standardization, and distribution of a full-featured, production quality library for reliable scientific
computing with routines using interval techniques for use by the wide community of applications developers.

It is known that there are feasible algorithms for minimizing convex functions, and that for general functions, global minimization is a difficult (NP-hard) problem. It is reasonable to ask whether there exists a class of functions that is larger than the class of all convex functions for which we can still solve the corresponding minimization prob...

Many constraint propagation techniques iterate through the constraints in a straightforward manner, but can fail because they do not take account of the coupling between the constraints. However, some methods of taking account of this coupling are local in nature, and fail if the initial search region is too large. We put into perspective newer met...

Based on work originating in the early 1970s, a number of recent global optimization algorithms have relied on replacing an original nonconvex nonlinear program by convex or linear relaxations. Such linear relaxations can be generated automatically through an automatic differ- entiation process. This process decomposes the objective and constraints...

During branch and bound search in deterministic global optimization, adaptive subdivision is used to produce subregions x, which are then eliminated, shown to contain an optimal point, reduced in size, or further subdivided. The various techniques used to reduce or eliminate a subregion x determine the efficiency and practicality of the algorithm....

An orthogonal basis for the null space of a rectangular m by n matrix, with m < n, is required in various contexts, and numerous well-known techniques, such as QR factorizations or singular value decompositions, are efiective at obtaining numerical approximations to such a basis. However, validated bounds on the components of each of these null spa...

A standard for the notation of the most used quantities and oper- ators in interval analysis is proposed.

An environment for general research into and prototyping of algorithms for reliable constrained and unconstrained global nonlinear optimization and reliable enclosure of all roots of nonlinear systems of equations, with or without inequality constraints, is being developed.

In this paper, we report a Fortran 90/95 software package, ParaGlobSol, that reliably finds numerical solutions for continuous nonlinear global optimization problems in parallel.

It is known that there are feasible algorithms for minimizing convex functions, and that for general functions, global minimization is a difficult (NP-hard) problem. It is reasonable to ask whether there exists a class of functions that is larger than the class of all convex functions for which we can still solve the corresponding minimization prob...

Many practical optimization problems are nonsmooth, and derivative-type methods cannot be applied. To overcome this difficulty, there are different concepts to replace the derivative of a function f :
: interval slopes, semigradients, generalized gradients, and slant derivatives are some examples. These approaches generalize the success of convex...

It is known that, in general, no computational techniques can verify the existence of a singular solution of the nonlinear system of n equations in n variables within a given region of n-space. However, computational veriÞcation that a given number of true solutions exist within a region in complex space containing is possible. That can be done by...

tic in practical algorithms involving interval Newton methods and other acceleration devices, in addition to the heuristic's e#ect within the simple "Moore--Skelboe" algorithm. The numerical experiments are carefully reported. With "Estimating and Validating the Cumulative Distribution of a Function of Random Variables: Toward the Development of Di...

Traditional computational fixed point theorems, such as the Kantorovich theorem (made rigorous with directed roundings), Krawczyk's method, or interval Newton methods use a computer's floating-point hardware computations to mathematically prove existence and uniqueness of a solution to a nonlinear system of equations within a given region of n-spac...

"This workshop focuses on complete solving techniques for continuous constraint satisfaction and optimization problems that provide all solutions with full rigor. Less rigorous solu- tion techniques are not excluded, since they may be part of complete relevant techniques. Complete solution techniques guarantee that all the constraints - e.g. securi...

As interval analysis-based reliable computations find wider application, more software is becoming available. Simultaneously, the applications for which this software is designed are becoming more diverse. Because of this, the software itself takes diverse forms, ranging from libraries for application development to fully interactive systems. The t...

Finding approximate solutions to systems of it nonlinear equations in it real variables is a much studied problem in numerical analysis. Somewhat more recently, researchers have developed numerical methods to provide mathematically rigorous error bounds on such solutions. (We say that we "verify" existence of the solution within those bounds on the...

Traditional computational fixed point theorems, such as the Kantorovich theorem (made rigorous with directed roundings), Krawczyk's method, or interval Newton methods use a computer's floating-point hardware computations to mathematically prove existence and uniqueness of a solution to a nonlinear system of equations within a given region of n-spac...

Deterministic global optimization with interval analysis involves
• using interval enclosures for ranges of the constraints, objective, and gradient to reject infeasible regions, regions without global optima, and regions without critical points;
• using interval Newton methods to converge on optimum-containing regions and to verify global optima....

this report wishes to thank the organizers, Christian Bliek and Djamila Sam-Harold for the administrative work that went into the workshop, and to thank the entire scientific committee for the excellent combination of topics and researchers

The GlobSol software package combines various ideas from interval analysis, automatic difierentiation, and constraint propagation to provide verifled solutions to unconstrained and constrained global op- timization problems. After brie∞y reviewing some of these techniques and GlobSol's development history, we provide the flrst overall description o...

Validated Computing 2002 took place in Toronto on May 23 to May 25, 2002, immediately after the Seventh SIAM Conference on Optimization and immediately before an informal workshop on vali-dated optimization at the Fields Institute on the University of Toronto Campus. Highlights included − a tutorial introduction to interval techniques, in the eveni...

Given an approximate solution to a nonlinear system of equations at which the Jacobi matrix is nonsingular, and given that the Jacobi matrix is continuous in a region about this approximate solution, a small box can be constructed about the approximate solution in which interval Newton methods can verify existence and uniqueness of an actual soluti...

Given an approximate solution to a nonlinear system of equations at which the Jacobi matrix is nonsingular, and given that the Jacobi matrix is continuous in a region about this approximate solution, a small box can be constructed about the approximate solution in which interval Newton methods can verify existence and uniqueness of an actual soluti...

Deterministic global optimization with interval analysis involves using interval enclosures for ranges of the constraints, objective, and gradient to reject infeasible regions, regions without global optima, and regions without critical points; using interval Newton methods to converge on optimum-containing regions and to verify global optima.

In this paper, we report a Fortran 90/95 software package, ParaGlobSol, that reliably flnds numerical solutions for continuous nonlinear global optimization problems in parallel. With this package, we have successfully solved some computational intensive real application problems. Superlinear speedup for some application have been observed because...

The reliable solution of nonlinear parameter estimation problems is an important computational problem in chemical engineering. Classical solution methods for these problems are local methods, and may not be reliable to nd the global optimum. Interval arithmetic can be used to compute completely reliably the global optimum for the nonlinear paramet...

Certain practical constrained global optimization problems have to date defied practical solution with interval branch and
bound methods. The exact mechanism causing the difficulty has been difficult to pinpoint. Here, an example is given where
the equality constraint set has higher-order singularities and degenerate manifolds of singularities on t...

Deterministic global optimization with interval analysis involves using interval enclosures for ranges of the constraints, objective, and gradient to reject infeasible regions, regions without global optima, and regions without critical points; using interval Newton methods to converge on optimum-containing regions and to verify global optima.

The Cluster Problem in Global Optimization the Univariate Case. We consider a branch and bound method for enclosing all global minimizers of a nonlinearC
2 or C
1 objective function. In particular, we consider bounds obtained with interval arithmetic, along with the “midpoint test,” but no acceleration procedures. Unless the lower bound is exact, t...

Computational fixed point theorems can be used to automatically verify existence and uniqueness of a solution to a nonlinear system of n equations in n variables ranging within a given region of n-space. Such computations succeed, however, only when the Jacobi matrix is nonsingular everywhere in this region. However, in problems such as bifurcation...

Computational fixed point theorems can be used to automatically verify existence and uniqueness of a solution to a nonlinear system of equations F (x) = 0, F : R n ! R n within a given region x of n-space. But such computations succeed only when the Jacobi matrix F 0 (x) is nonsingular everywhere in x. However, in many practical problems, the Jacob...

The algorithm contains a moderately-sized system of Fortran-90 subroutines, along with a driver program. The output of this system is a Fortran-77 program for evaluating the derivative of a user-specified function f : R m ! R n , where m and n are arbitrary. The user defines f as a Fortran-90 subroutine, with certain syntax restrictions. The statem...

Interval branch and bound algorithms for finding all roots use a combination of a computational existence / uniqueness procedure and a tesselation process (generalized bisection). Such algorithms identify, with mathematical rigor, a set of boxes that contains unique roots and a second set within which all remaining roots must lie. Though each root...

Techniques for verifying feasibility of equality constraints are presented. The underlying verification procedures are similar to a proposed algorithm of Hansen, but various possibilities, as well as additional procedures for handling bound constraints, are investigated. The overall scheme differs from some algorithms in that it rigorously verifies...

Various algorithms can compute approximate feasible points or approximate solutions to equality and bound constrained optimization problems. In exhaustive search algorithms for global optimizers and other contexts, it is of interest to construct bounds around such approximate feasible points, then to verify (computationally but rigorously) that an...

this paper. First, required properties of interval extensions for non-smooth functions are discussed. Then, selected formulas from [4] are presented. The formulas for slopes presented here represent improvements (sharper bounds) over those in [4]. A simple, illustrative example is then given. Fourth, a convergence and existence / uniqueness verific...

It is known that, in general, no computational techniques can verify the existence of a singular solution of the nonlinear system of n equations in n variables within a given region x of n-space. However, computational verication that a given number of true solutions exist within a region in complex space containing x is possible. That can be done...

We consider branch and bound methods for enclosing all unconstrained global minimizers of a nonconvex nonlinear twice-continuously differentiable objective function. In particular, we consider bounds obtained with interval arithmetic, with the "midpoint test," but no acceleration procedures. Unless the lower bound is exact, the algorithm without ac...

Certain cases in which the interval hull of a system of linear interval equations can be computed inexpensively are outlined. We extend a proposed technique of Hansen and Rohn with a formula that bounds the solution set of a system of equations whose coefficient matrix A = [A; A] is an H-matrix; when A is centered about a diagonal matrix, these bou...

Interval Newton methods in conjunction with generalized bisection can form the basis of algorithms that find all real roots within a specified box X ae R n of a system of nonlinear equations F (X) = 0 with mathematical certainty, even in finite-precision arithmetic. In such methods, the system F (X) = 0 is transformed into a linear interval system...

this paper, we will concentrate on optimal preconditioners, computed row-by-row, only. A preconditioned interval Gauss--Seidel method may be used to compute a new interval ~ x k for the k-th variable. Suppose Y k = (y k1 ; y k2 ; :::; y kn ) is the preconditioner for x k . Algorithm 1 (Preconditioned Gauss--Seidel method) 1. Compute Y k F 0 Delta (...

Most interval branch and bound methods for nonlinear algebraic systems have to date been based on implicit underlying assumptions of continuity of derivatives. In particular, much of the theory of interval Newton methods is based on this assumption. However, derivative continuity is not necessary to obtain effective bounds on the range of such func...

A rigorous and efficient algorithm is presented for computing a sequence of points on all the branches of surface patch intersection curves within a given box. In the algorithm, an interval step control continuation method makes certain that the predictor algorithm will not jump from one branch to the another. These reliability properties are indep...

It has been known how to use computational xed point theorems to verify existence and uniqueness of a true solution to a nonlinear system of equations within a small region about an approximate solution. This can be done in O n 3 operations, where n is the number of equations and unknowns. However, these standard techniques are only valid if the Ja...

Many practical optimization problems are nonsmooth, and derivative-type methodscannot be applied. To overcome this diculty, there are dierent approaches to replace the derivativeof a function f : Rn! R: interval slopes, semigradients, generalized gradients, and slant derivativesare some examples. In this paper we study the relationships among these...

Certain practical constrained global optimization problems have to date defied practical solution with interval branch and bound methods. The exact mechanism causing the difficulty has been difficult to pinpoint. Here, an example is given where the equality constraint set has higher-order singularities and degenerate manifolds of singularities on t...

Traditionally, iterative methods for nonlinear systems use heuristic domain and range stopping criteria to determine when accuracy tolerances have been met. However, such heuristics can cause stopping at points far from actual solutions, and can be unreliable due to the effects of round-off error or inaccuracies in data. In verified computations, r...

Deterministic global optimization requires a global search with rejection of subregions. To reject a subregion, bounds on the range of the constraints and objective function can be used. Although sometimes eective, simple interval arithmetic sometimes gives impractically large bounds on the ranges. However, Taylor models as developed by Berz et al...

We apply interval techniques for global optimization to several industrial applications including Swiss Bank (currency trading), BancOne (portfolio management), MacNeal-Schwendler (Þnite element), GE Medical Systems (Magnetic resonance imaging), Genome Theraputics (gene prediction), inexact greatest common divisor computations from computer algebra...

. Interval iteration can be used, in conjunction with other techniques, for rigorously bounding all solutions to a nonlinear system of equations within a given region, or for verifying approximate solutions. However, because of overestimation which occurs when the interval Jacobian matrix is accumulated and applied, straightforward linearization of...