4.1. Nonlinear Interval Uncertain Optimization Problem
The nonlinear interval uncertain optimization can be expressed as follows [
16]:
where
X is the n-dimensional design vector whose value range is
.
U is a q-dimensional uncertain vector, which is described by a q-dimensional interval vector
.
f and
are objective function and the
i-th constraint, respectively. For the optimization model of the gun engraving system, the engraving resistance force
is taken as an objective function. Because
can affect the engraving-completion velocity of the projectile. Moreover, the engraving-completion velocity of the projectile is the most crucial index for designers. Additionally, the temperature of the propellant gas
T can affect the erosion degree and service life of the gun barrel. The pressure wave of the propellant gas
can affect the propellant combustion stability. Both
T and
should be in a feasible scope. Therefore,
T and
are taken as a constraint.
l is the number of constraints. They are all continuous nonlinear functions of
X and
U, and at least one of them is a nonlinear function of
U.
is the allowable interval of the
i-th uncertain constraint, and it can also be a real number in practical problems.
Since the objective function and the constraint are both continuous functions of U, for any design variable X that is determined, the values of the objective function and the constraint are also intervals.
4.2. Affine Arithmetic
In interval arithmetic, two interval numbers are set as
,
, and the basic operation rules are as follows:
The interval arithmetic can directly calculate the interval number. Although the calculation is simple, it will bring about the interval expansion phenomenon. When calculating the function value range, the calculation result interval is amplified. For example, calculate , where . The resulting interval obtained from Equation (19) is , while the actual result is . The root of the interval expansion problem comes from the correlation between interval numbers.
The affine arithmetic is similar to interval arithmetic, but the affine arithmetic considers the correlation of variables in the calculation, so it can well suppress the interval expansion. Moreover, when the same variable appears in different terms of the function, the affine arithmetic can retain its complete correlation to the greatest extent, while the interval arithmetic will cause serious information loss. The affine form of the uncertain quantity
is denoted as
:
where
is the central value of the affine afform, and the
are noise symbols, whose values are unknown but definitely lie in the interval [−1, 1], correspondingly,
are the known real coefficients of the noise symbols. When
, the affine form obtains its maximum and minimum values, which can be expressed as the interval:
For interval uncertain optimization problems, the uncertain interval quantity can be expressed as:
where
and
are the central value and radius of the interval, respectively.
Affine afform
and
:
,
The basic affine operation rules are as follows:
The quadratic term
appears in the multiplication operation.
can be obtained from
, so the upper and lower bounds of the affine operation
can be expressed as interval
, where:
4.3. An Optimization Method Based on Affine Arithmetic
In this work, a nonlinear interval uncertain optimization method based on affine arithmetic is adopted, and the phrase “affine arithmetic-based method” is used for calling this method. First, the uncertain variables in the objective function and constraint are rewritten into the affine form. Second, the affine arithmetic is utilized to obtain the interval of objective function and constraint of the design variable X, and then the uncertain problem is transformed into a deterministic problem. Finally, the genetic algorithm is used to solve the problem.
The details of the approach are summarized below.
First, let
. Then, the objective function is rewritten into affine form.
where
is the central value of the affine form,
are the known real coefficient,
are the noise symbol. Therefore, the interval of the objective function at the design variable
can be obtained:
Second, the constraints are rewritten in affine form:
where
is the central value of the affine form,
are the known real coefficient,
are the noise symbol. The interval of the constraint
at the design variable
can be obtained:
The above uncertain problem can be rewritten as:
It can be further rewritten as:
At this point, the uncertain optimization problem is transformed into a single-layer deterministic optimization problem, which can be solved by existing sequential quadratic programming (SQP), genetic algorithm (GA) and other intelligent optimization algorithms. This method is further illustrated with a numerical example below.
This numerical example is a nonlinear interval optimization problem with three uncertain variables, and it is from a doctoral thesis [
34]. In addition, all the parameters, which we set, are the same as the parameters in reference [
34].
Step 1: rewriting the interval quantity into affine quantity.
Let ,,.
Step 2: computing the affine form of the objective function.
The nonlinear term of the uncertain variables in the objective function are
and
:
The objective function can be rewritten as:
Since
, the value interval of even term
are
. The interval of the objective function can be obtained as:
Step 3: computing the affine form of the constraints.
The nonlinear term of the uncertain variables in the objective function are
and
:
The constraint
can be rewritten as:
Since
, the value interval of even term
is
. The interval of constraint
can be obtained as:
Similarly, the affine form of the constraint
can be obtained as:
Step 4: deterministic transformation.
The original uncertain optimization problem can be rewritten as the optimization problem in the following affine form:
According to the interval order model proposed in Reference [
16], the above optimization problem can be transformed into the following deterministic optimization problem:
So far, the nonlinear interval uncertainty optimization problem has been transformed into a deterministic optimization problem. The linear weighting method can be used to further convert the problem into a single-objective optimization problem:
where
is the multiobjective evaluation function,
is the multiobjective weight coefficient,
is the parameter that guarantees
and
to be nonnegative,
and
are regularization factors of a multiobjective function.
In order to be consistent with reference [
34], the regularization factors
,
and
are 1.33, 0.34 and 0.0, respectively. GA [
35] is used to solve the problem, and the results are listed in
Table 5.
The calculation results obtained by the two-layer nested method are regarded as the reference solution for interval uncertain optimization problem. Comparing with the results by these two methods, the max error is only 1.6%. It indicates that the affine arithmetic-based method has high accuracy. Furthermore, the affine arithmetic-based method costs only 1.944 s to complete the numerical example, while the two-layer nested method costs 544.48 s. It also indicates that the affine arithmetic-based method has obvious improvement in computational efficiency.