LP Narrowing: A New Strategy for Finding All Solutions of Nonlinear Equations Kiyotaka Yamamura Naoya Tamura Koki Suda Chuo University, Tokyo, Japan

Introduction ﻪIn this presentation, we discuss the problem of finding all solutions of a system of n nonlinear equations with a separable mapping: f (x) = 0 (1) contained in a box D in R n. ﻪActually, the algorithm proposed in this presentation can be applied to more general systems of nonlinear equations, but we restrict our discussion to the separable systems because the proposed algorithm is especially efficient for such systems. In this presentation, we do not consider verified numerical computation because we mainly consider the application to large- scale practical engineering problems where it is enough to obtain approximate solutions. Finding all solutions of nonlinear equations is an important problem which is widely encountered in science and engineering.

Interval Algorithms ﻪAs a computational method to find all solutions of nonlinear equations, interval analysis based techniques are well-known. ﻪUsing the interval algorithms, all solutions of (1) contained in D can be found with mathematical certainty. ﻪHowever, the computation time of the interval algori- thms tends to grow exponentially with the dimension n. ﻪTherefore, it is necessary to develop a powerful test for nonexistence of a solution in a given box.

LP Test ﻪIn [1], a powerful computational test was proposed for nonexistence of a solution to the system of nonlinear equations (1) in a given box X. ﻪIn [2], the LP test was much improved by introducing the dual simplex method, by which the LP test becomes not only powerful but also efficient. In [2], the Krawczyk-Moore algorithm using the LP test succeeded in finding all solutions to systems of nonlinear equations with n = 200. ﻪIn [3], an improved version of this algorithm is proposed, which succeeded in finding all solutions of systems of nonlinear equations with n = 300.

Purpose of the Study We propose an efficient algorithm for finding all solutions of nonlinear equations using a new strategy called LP narrowing. Boxes containing no solution are excluded. Boxes containing solution are narrowed. It is shown that the proposed algorithm could find all solutions of systems of 5000 － nonlinear equations in practical computation time.

Basic Procedure of Interval Algorithms An n-dimensional interval vector is denoted by If there is no solution of (1) in X, then we exclude it from further consideration. If there is a unique solution of (1) in X, then we compute it by some iterative method. If these conditions are not satisfied, then bisect X to form two new boxes; we then continue the above procedure with one of these boxes, and put the other one on a stack for later consideration. X In interval algorithms, the following procedure is performed recursively, beginning with the initial box X = D. (At each level, we analyze the box X.) Geometrically, X is an n-dimensional box. Thus, we can find all solutions of (1) contained in D with mathematical certainty.

Nonexistence test proposed in [1] (2) For the simplicity of notation, and without loss of generality, in this presentation we assume that (1) can be represented as

LP Test (2) (3) Let the interval extension of g i (x i ) over [a i, b i ] be [c i, d i ]. Then, we introduce y i and put y i = g i (x i ). Now we replace each nonlinear function g i (x i ) in (2) by y i, and consider the LP problem (3). Then, we apply the simplex method to (3). X ＝（［ a 1, b 1 ］, …, ［ a n, b n ］）

LP Test Using the Simplex Method ﻪBy introducing the LP test to the interval algorithms, all solutions of (2) can be found very efficiently. ﻪIn [1], this algorithm solves a system of nonlinear equations with n = 60 in practical computation time, although the original Krawczyk-Moore algorithm can solve the system only for n ＜ 12. ・ All solutions of (2) that exist in X satisfy the constraints in (3). ・ If the LP problem (3) is not feasible, then we can conclude that there is no solution of (2) in X. ・ The feasibility of (3) can be checked by the simplex method. (LP test)

LP Test Using the Dual Simplex Method ﻪIn [2], it is shown that the LP test can be performed with a few iterations (often no iteration) per box by using the dual simplex method. ﻪUsing this technique, the LP test becomes not only powerful but also efficient. ﻪIn [2], this improved LP test is introduced to the Krawczyk-Moore algorithm, which could find all solutions of systems of nonlinear equations with n = 200. ﻪIn [3], an improved version of this algorithm is proposed, which succeeded in finding all solutions of systems of nonlinear equations with n = 300.

Proposed Algorithm ﻪThe proposed algorithm is an extension of the algorithm in [2], to which the idea of narrowing a box using LP techniques is introduced. ﻪIf X is not excluded, then we narrow the box so that no solution is lost, which makes the algorithm much more efficient. ﻪNow we explain how X is narrowed efficiently by using the LP techniques.

If the feasible region of (4) is empty, then we exclude X from further consideration. from lower side Narrowing from lower side in x i - direction x2x2 x1x1 First, we apply the dual simplex method to (4) for i = 1. X （4）（4） If the minimum value x 1 * is greater than a 1, then we prune the lower part of X.

from lower side Narrowing from lower side in x i - direction Then, we repeat the similar narrowing procedure in the x i - directions (i > 1), and narrow the box in all coordinate directions from the lower sides. x2x2 x1x1 X

from upper side Narrowing from upper side in x i - direction Then, we apply the dual simplex method to (5) for i = 1. （5）（5） x2x2 x1x1 X If the maximum value x 1 * is less than b 1, then we prune the upper part of X. If the feasible region of (5) is empty, then we exclude X from further consideration.

from upper side Narrowing from upper side in x i - direction x2x2 x1x1 X Then, we repeat the similar narrowing procedure in the x i - directions (i > 1), and narrow the box in all coordinate directions from the upper sides. Such a series of procedures is called LP narrowing.

LP narrowing ﻪAs the box becomes smaller, the feasible region becomes smaller, which makes the LP narrowing more and more powerful. ﻪThe LP problem (4) or (5) can be solved efficiently with a few iterations by the dual simplex method. The LP narrowing is not only powerful but also efficient and narrows a box very effectively. ﻪNotice that, in the LP narrowing, we first per-form the narrowing procedure from the lower sides of all x i -directions, and then perform the narrowing procedure from the upper sides. ﻪThis is because it makes the number of iteration in the dual simplex method small and makes the algorithm efficient.

Implementation of the Proposed Algorithm 1 The proposed algorithm can be easily implemented by using the free package GLPK (GNU Linear Programming Kit ) 1. ﻪCallable library for C ﻪIntended to solve large-scale LP problems ﻪKnown to be very efficient; in many cases, it is faster and more robust than lp_solve 5.5. ﻪWork in progress and presently under continual development ﻪAs of the current version 4.27, it is able to handle problems with up to constraints.

Advantages of Using GLPK GLPK is not only very efficient but also well-suited to the proposed algorithm. ﻪSince the bounded-variable technique is implemented in GLPK, it can solve the LP problems of the form (4) or (5) very efficiently. ﻪWe can easily perform the dual simplex method starting from a previously obtained dual feasible basis by using the control parameter “GLP_DUALP”.

Numerical Examples ﻪ Programming language: C (double precision) ﻪ Dell Precision T7400 (Intel Xeon 3.4GHz) ﻪ We used GLPK for solving the LP problems. ﻪ We compare the computation time of the proposed algorithm and the algorithm proposed in [3].

Example 1 A system of n nonlinear equations (known as Yamamura1) Initial Region:

Comparison of computation time (s) in Example 1nSRef.[3]Proposed : :: : ∞ ∞ ∞ ∞ ∞ minutes 34 hours S denotes the number of solutions obtained by the algorithms. ∞ denotes that it could not be computed in practical computation time.nSBoxesProposed : :: : “Boxes” denotes the number of analyzed boxes of the proposed algorithm. It is also seen that the number of analyzed boxes is very small in the proposed algorithm, which implies that the LP narrowing is very powerful. Notice that the number of analyzed boxes does not become large as n increases; it depends mainly on the number of solutions.

Example 2 A system of n nonlinear equations Initial Region:

Comparison of computation time (s) in Example 2nSRef.[3]Proposed : :: : 10003∞ ∞ ∞ ∞ ∞ The total number of pivotings 42,445 The average number of pivotings in solving the LP problem 0.47 It is seen that a similar result is obtained as that in Example 1. Considering the size of the problem, this number is small.

Example 3 Initial Region: A system of n nonlinear equations This system comes from a nonlinear two-point boundary value problem termed the Bratu problem.

nSRef.[3]Proposed : :: : 10002∞ ∞ ∞ ∞ ∞ ∞ ∞ Comparison of computation time (s) in Example 3 The total number of pivotings 117,326 The average number of pivotings in solving the LP problem 1.9 The number of solutions is two for all n.nSBoxesProposed : :: : hours 10 minutes The number of analyzed boxes of the proposed algorithm is only three for all n.

Proposed Algorithm (Example 3) ﻪThe average narrowing rate per direction (n = 10000) 0.59 ﻩin the first box: ﻩfrom the second box: It is seen that the LP narrowing is very powerful and narrows a box very rapidly, especially when the box contains one solution. ﻪThe number of analyzed boxes is only three for all n.

Proposed Algorithm (Example 3) x2x2 x1x1 Proposed Algorithm The proposed algorithm narrowed the boxes as above. This is the reason why the proposed algorithm is very efficient for this problem, and could solve the NP-hard problem for n = in practical computation time.

Example 4 Transistor Circuits 0.01 s 3 solutions 0.02 s 9 solutions 0.10 s 11 solutions 0.07 s one solution Systems of nonlinear equations containing many strongly nonlinear terms of the form exp(40x i - 1) It is seen that all solutions were found in little computation time.

RealPaver [4] ﻪRealPaver is a well-known interval software package for solving numerical constraint satisfaction problems including finding all solutions of nonlinear equations. ﻪCompare with four algorithms in RealPaver (called BC3N, BC5, weak3B, and 3B) which are considered to be the most efficient algorithms there. [4]

nSBC3NBC5weak3B3BProposed ∞0.013 nBC3NBC5weak3B3BProposed ∞0.009nBC3NBC5weak3B3BProposed Example 1. (sec) Example 2. (sec) Example 3. (sec) Comparison with RealPaver For these problems, the proposed algorithm is much more efficient than the algorithms in RealPaver.

Conclusion ﻪAn efficient algorithm has been proposed for finding all solutions of separable systems of nonlinear equations using a new strategy called LP narrowing. ﻪIt has been shown that the proposed algorithm is very efficient and has the possibility of solving large-scale systems of nonlinear equations in practical computation time. ﻪThe proposed algorithm can be easily implemented by using GLPK. The proposed algorithm seems to be a useful tool for finding all solutions of nonlinear equations. ﻪThe interesting feature of this algorithm is that the number of analyzed boxes is very small, although LP problems have to be solved 2n times for each box. The computational cost of solving LP problems 2n times seems to be very large, but actually they can be solved efficiently by the dual simplex method.