Disjoint Feasible Space
Problem Definition
Keywords
optimisation, mdo
Introduction
To optimise a system under certain operating conditions, means finding the best configuration of the system that minimises (or maximises) one (or more) performance parameters while satisfying a set of requirements.
Design variables (\(\mathbf{x}\))
Design variables (\(\mathbf{x}\)) are a set of (mostly) independent parameters that completely define a system configuration.
Design variables typically exhibit these characteristics, but they can also deviate from these norms. While they are often continuous, they can also be discrete. They are generally bounded, but unbounded instances exist. Most of the time, these variables are independent, yet dependent scenarios are possible. They are usually deterministic, but stochastic situations can occur. More often than not, they are static, but dynamic cases are not unheard of. The various combinations of these factors result in different kinds of optimization problems, each requiring a unique solution strategy.
For example, for a wing optimisation problem, we may choose the following design variables:
\[\mathbf{x}= \begin{bmatrix} \text{Area}\\ \text{Span}\\ \text{Sweep}\\ \text{Taper Ratio}\\ \text{Aspect Ratio}\\ \text{Twist}\\ \text{Dihedral}\\ \text{Incidence}\\ \text{Airfoil}\\ \text{Thickness}\\ \text{Camber}\\ \end{bmatrix}\]
We make following observations.
The proposed set of design variables are not independent. The wing area and the wing span are related by the wing aspect ratio. Given any two of these Design variables, the third is fixed. Hence this is a redundant set of design variables. Hence, wing span should not be chosen as a design variable. Design variables should be indenpendent of each other.
Choice of design variables is dependent on the objective function. For example, the given set of design variables is suitable for a wing optimisation problem where the objective function is to maximise the lift-to-drag ratio. However, this statement comes with a caveat. If the flow regime is low subsonic, then the sweep is not a relevant design variable since it will not affect the objective function much. On the other hand, if the flow regime is supersonic, then the thickness is not a relevant design variable since we already know that the optimum thickness is zero.
Choice of design variables is not unique. For example, the wing area and the wing aspect ratio coulb have been replaced with the wing span and the wing aspect ratio.
All the design variables are not continuous. For example, the airfoil is a discrete variable. The airfoil is chosen from a set of discrete airfoils. This will severely limits the search space. Also it will affect the choice of the optimisation algorithm and consequently the computational cost of the optimisation process. Choice and number of design variables affects the computational cost of the optimisation process.
Current set of design variables explore in detail the geometry of the wing however limit the design space to a finite number of airfoils. In other words, the designer is betting on the fact that he can make an educated guess on the kind of airfoils that would maximise \(C_L/C_D\). This shows the implicit assumptions behind the problem statement. Hence, every choice of a set of design variables reflects the implicit assumptions about the problem statement.
The choice of design variables also reflects the area of the design space that the designer is interested in exploring. For example, the designer may be interested in exploring the effect of sweep on the \(C_L/C_D\) for a given airfoil. In that case, the design variables will be chosen as follows:
\[\mathbf{x}= \begin{bmatrix} \text{Sweep}\\ \text{Airfoil}\\ \end{bmatrix}\]
As a thumb-rule, the less you know about the system and the more you want to explore the design space, the more the number of design variables. However, the more the number of design variables, the more the computational cost of the optimisation process. In some cases, a large optimisation problem may not be even solveable. Hence, choice of design variables is a trade-off between the computational cost and the exploration of the design space.
The correct choice of design variables is critical to the success of the optimisation process. An optimiser aims to capture the largest possible design space using the smallest possible number of design variables. However, if the design space is too large, then the solution process becomes too expensive. If the design space is too small, then the solution process may not be able to find the optimum.
Design Parameters (\(\mathbf{p}\))
The operating conditions are defined by a set of parameters that define the environment in which the system operates. These parameters can also present the approximations and assumptions under which the problem is framed. For example, the operating conditions for a wing optimisation problem are as follows:
\[\begin{bmatrix} \text{Cruise Mach Number}\\ \text{Cruise Angle of Attack}\\ \text{Altitude}\\ \end{bmatrix}\]
As we know, \(C_D \doteq C_D(M, Re)\), where \(M\) is the Mach number and \(Re\) is the Reynolds number. In order to calculate the Reynolds number, the variation of temperature, viscosity and density with the altitude must be known. Hence, these parameters are to be supplemented by the atmospheric model (like ISA) to be complete.
It should be noted here that a design variable can always be converted to a design parameter thereby reducing the design space. For example, we could maximise \(C_L/C_D\) for a specific airfoil. The name of the specific airfoil will get added to the design parameter vector whereas the dimensionality of the design space will reduce by one.
Objective Function \(f(\mathbf{x})\)
The performance parameters are called the objective function (\(f(\mathbf{x})\)).
Constraints \(g(\mathbf{x}), h(\mathbf{x})\)
The requirements are called the constraints (\(g(\mathbf{x}), h(\mathbf{x})\)). There are two types of constraints:
Inequality constraints: These are constraints that must be satisfied. For example, the wing area must be greater than zero. These constraints are written as follows:
\[g(\mathbf{x}) \leq 0\]
Equality constraints: These are constraints that must be satisfied. For
\[h(\mathbf{x}) = 0\]
For example, \(C_L = C_{L_\text{cruise}}\) is an equality constraint while \(\lambda \geq 0.7\) is an inequality constraint. This constraint may be necessary to ensure that the wing is structurally stable. Calculation of constraint at a design point is as difficult as the objective function. As a thumb-rule, equality constraints are more difficult to satisfy than inequality constraints. Equality constraints generally reduce the dimensionality of the design space, while inequality constraints exclude certain regions of the design space.
To visualise this, consider following problem:
\[\begin{aligned} \min_{x, y} x^2 + y^2\\ \text{subject to } x &\geq 0\\ y &\geq 0\\ x &= 1\\ x - y &> 0\\ \end{aligned}\]
This is clearly a two-dimensional problem. However, the equality constraint reduce the dimensionality of the problem to one. On the other hand, the inequality constraint excludes the region \(x<0\).
Multiple Plots
To understand the above plots more clearly, let us first define a few terms.
- Design space
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The n-dimensional space spanned by the design variables is called the design space. If the design variables are independent of each other, then it is a n-dimensional hypercube where $^n.
- Feasible Space
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The feasible space is the subset of the design space that satisfies all the constraints. Each point in the feasible space corresponds to a viable realisable system that satisfies all the constraints. Remember that feasbile space may not be a continuous region. It may be a set of disjointed regions in the design space.
\[\begin{aligned} \min_{x, y} \quad f(x,y) &= x^2 + y^2\\ \text{subject to } y &\geq 0.36\\ y - x^2 &\leq 0\\ \end{aligned}\]
The feasible space is the region bounded by a line and the parabola. It is a disjointed region. The optima is at \((0, 0.36)\) and \((0, -0.36)\).
Optimisation process aims to find optimal configuration (\(\mathbf{x}^*\)) that minimises the objective function \(f(\mathbf{x}^*)\) while satisfying the constraints (inequatlity \(g(\mathbf{x}^*)\leq 0\), equality \(h(\mathbf{x}^*)=0\)).
In order to standardise the process of optimisation, and let researchers talk to each other about their work, a standard notation has been developed. The notation is not perfect, but it is widely used.
You may find some references that deviate from the standard problem formulation. For example, some authors develop their analysis for maximisation of the objective function. In that case, the necessary and sufficient conditions (and the KKT conditions) will be different.
The notation is as follows:
\[\begin{aligned} \min_{\mathbf{x}} f(\mathbf{x})\\ \text{subject to } g(\mathbf{x}) &\leq 0\\ h(\mathbf{x}) &= 0\\ \mathbf{x}_{\text{min}} &\leq \mathbf{x}\leq \mathbf{x}_{\text{max}} \end{aligned}\]
where \(\mathbf{x}\) is the design vector, \(f(\mathbf{x})\) is the objective function, \(g(\mathbf{x})\) is the inequality constraint, \(h(\mathbf{x})\) is the equality constraint, and \(\mathbf{x}_{\text{min}}\) and \(\mathbf{x}_{\text{max}}\) are the lower and upper bounds on the design vector.
Most optimisation problems can be converted to this standard form.
Arora, Jasbir. 2012. Introduction to Optimum Design. Academic Press.
Here are some examples of how to convert non-standard problems to standard form.
\(g(x) \geq 0\) can be converted to \(-g(x) \leq 0\)
\(g(x) < 0\) might look different. But it is essentially the same as \(g(x) \leq 0\) since all the numerical algorithms work in tolerances for constraint satisfaction. If we wish ensure that this constrain is satisfied to certain tolerance, then we can pose if as \(g(x) + \epsilon \leq 0\) where \(\epsilon\) can be set to the desired value.
\(g(x) > 0\) can be posed as \(-(g(x) + \epsilon) \leq 0\)
\(h(x) \neq 0\)
This is a rare form of constraint which is extremely rare. It can be posed as \(- h(x)^2 \leq 0\). After optimisation, if the constraint is active, then the optimum can be perturbed a little bit. Strictly speaking, as all optimisation algorithms satisfy constraints within a tolerance, we do not really have g(x)=0 even if the constraint is active. If it is needed to ensure that \(| h(x) | > \epsilon\), where \(\epsilon\) is some distance that you want to impose from \(0\) then set \(h(x) + \epsilon \geq 0\quad \mbox{and} \quad h(x) - \epsilon \leq 0\)