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LP Narrowing: A New Strategy for Finding All Solutions of Nonlinear Equations

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LP Narrowing: A New Strategy for Finding All Solutions of Nonlinear Equations

Kiyotaka Yamamura Naoya Tamura Koki Suda

Chuo University, Tokyo, Japan

In 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 Rn.

Actually, 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.

Finding all solutions of nonlinear equations is an important problem which is widely encountered in science and engineering.

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.

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

- In [1], a powerful computational test was proposed for nonexistence of a solution to the system of nonlinear equations (1) in a given box X.
- In [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.
- In [3], an improved version of this algorithm is proposed, which succeeded in finding all solutions of systems of nonlinear equations with n = 300.

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 5000－50000 nonlinear equations in practical computation time.

X

An n-dimensional interval vector is denoted by

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.

- 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 bisectX 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.

Thus, we can find all solutions of (1) contained in D with mathematical certainty.

(2)

For the simplicity of notation, and without loss of generality, in this presentation we assume that (1) can be represented as

(2)

X＝（［a1, b1］, … ,［an, bn］）

(3)

Let the interval extension of gi(xi) over [ai , bi] be [ci , di].

Then, we introduce yi and put yi = gi(xi).

Now we replace each nonlinear function gi(xi) in (2) by yi, and consider the LP problem (3).

Then, we apply the simplex method to (3).

・ 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 bythe simplex method.

- By introducing the LP test to the interval algorithms, all solutions of (2) can be found very efficiently.
- In [1], this algorithm solves a system of nonlinear equations withn = 60in practical computation time, although the originalKrawczyk-Moore algorithmcan solve the system only forn＜ 12.

(LP test)

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

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

If the minimum value x1* is greater than a1, then we prune the lower part of X.

x2

First, we apply the dual simplex method to (4) for i = 1.

X

（4）

x1

If the feasible region of (4) is empty, then we excludeX from further consideration.

x2

X

Then, we repeat the similar narrowing procedure in the xi - directions (i > 1), and narrow the box in all coordinate directionsfrom the lower sides.

x1

If the maximum value x1* is

less than b1, then we prune

the upper part of X.

x2

Then, we apply the dual simplex method to (5) for i = 1.

X

（5）

x1

If the feasible region of (5) is empty, then we excludeX from further consideration.

x2

Then, we repeat the similar narrowing procedure in the xi - directions (i > 1), and narrow the box inall coordinate directionsfrom the upper sides.

X

Such a series of procedures is calledLP narrowing.

x1

- As the box becomes smaller, the feasible region becomes smaller, which makes the LP narrowingmore and more powerful.
- The LP problem (4) or (5) can be solved efficiently with a few iterations by the dual simplex method.

- Notice that, in the LP narrowing, we first per-form the narrowing procedure from the lower sides ofall xi-directions, and then perform the narrowing procedure from the upper sides.
- This is because it makes the number of iteration in the dual simplex method small and makes the algorithm efficient.

The LP narrowing is not only powerful but also

efficient and narrows a box very effectively.

The proposed algorithm can be easily implemented by using the free package GLPK(GNULinear ProgrammingKit )1.

- Callable library for C
- Intended to solve large-scale LP problems
- Known to be very efficient; in many cases, it is faster and more robust than lp_solve 5.5.
- Work in progress and presently under continual development
- As of the current version 4.27, it is able to handle problems with up to 100 000 constraints.

1http://www.gnu.org/software/glpk/

- We can easily perform the dual simplex method starting from a previously obtained dual feasible basis by using the control parameter “GLP_DUALP”.

- Since the bounded-variable technique is implemented in GLPK, it can solve the LP problems of the form (4) or (5) very efficiently.

GLPKis not only very efficient but also well-suited to the proposed algorithm.

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

A system of n nonlinear equations

(known as Yamamura1)

Initial Region:

Comparison of computation time (s) in Example 1

“Boxes” denotes the number of analyzed boxes of the proposed algorithm.

S denotes the number of solutions obtained by the algorithms.

∞ denotes that it could not be computed in practical computation time.

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.

10 minutes

Notice that the number of analyzed boxes does not become large as n increases; itdepends mainly on the number of solutions.

34 hours

A system of n nonlinear equations

Initial Region:

Comparison of computation time (s) in Example 2

It is seen that a similar result is obtained as that in Example 1.

Considering the size of the problem, this number is small.

The total number of pivotings

42,445

The average number of pivotings

in solving the LP problem

0.47

A system of n nonlinear equations

Initial Region:

This system comes from a nonlinear two-point boundary value problem termed the Bratu problem.

Comparison of computation time (s) in Example 3

The number of solutions is two for all n.

The total number of pivotings

117,326

The average number of pivotings

in solving the LP problem

1.9

The number of analyzed boxes of the proposed algorithm is only three for all n .

10 minutes

8 hours

- The average narrowing rate per direction (n = 10000)
- in the first box: 0.59
- from the second box: 0.000072

It is seen that the LP narrowing is very powerful and narrows a box very rapidly, especially when the box contains one solution.

- The number of analyzed boxes is only three for all n.

x2

This is the reason why the proposed algorithm isvery

efficientfor this problem, and could solve the NP-hard

problem for n = 50000 in practical computation time.

x1

Proposed Algorithm

The proposed algorithm narrowed the boxes as above.

Example 4Transistor Circuits

Systems of nonlinear equations containing many strongly nonlinear terms of the form exp(40xi - 1)

0.02s

0.01 s

It is seen that all solutions were found in little computation time.

3 solutions

9 solutions

0.07 s

0.10s

one solution

11 solutions

RealPaver is a well-known interval software package for solving numerical constraint satisfaction problems including finding all solutions of nonlinear equations.

Compare with four algorithms in RealPaver (called BC3N, BC5, weak3B, and 3B) which are considered to be the most efficient algorithms there.

[4]

Comparison with RealPaver

Example 1. (sec)

For these problems, the proposed algorithm is much more efficient than the algorithms in RealPaver.

Example 2. (sec)

Example 3. (sec)

- An efficient algorithm has been proposed for finding all solutions of separable systems of nonlinear equations using a new strategy called LP narrowing.
- It 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.
- The proposed algorithm can be easily implemented by using GLPK.

- The 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.

The proposed algorithm seems to be a useful tool

for finding all solutions of nonlinear equations.