Problem Formulation

Properly formulating an optimisation problem is crucial for finding effective solutions. This page outlines the key steps and components of problem formulation.

Key Components

1. Objective Function ($L$)

The objective function is a mathematical expression that quantifies what we want to optimize:

• For minimization: Find ${\theta }^{\ast }$ such that $L({\theta }^{\ast })\le L(\theta )$ for all $\theta$
• For maximization: Find ${\theta }^{\ast }$ such that $L({\theta }^{\ast })\ge L(\theta )$ for all $\theta$

Examples of objective functions include:

• Cost functions to be minimized
• Profit functions to be maximized
• Error functions to be minimized
• Efficiency functions to be maximized

2. Design Vector ($\theta$)

The design vector consists of all variables that can be adjusted to optimize the objective function:

• $\theta \in {\mathbb{R}}^D$, where $D$ is the dimension (number of variables)
• Each component ${\theta }_i$ represents one design variable
• The design space is the set of all possible design vectors

3. Constraints

Constraints limit the feasible values of the design variables:

Equality constraints:

• $E_1(\theta )=0,E_2(\theta )=0,...,E_m(\theta )=0$

Inequality constraints:

• $I_1(\theta )\ge 0,I_2(\theta )\ge 0,...,I_n(\theta )\ge 0$

The feasible region is the set of all design vectors satisfying all constraints.

Mathematical Formulation

The general form of an optimisation problem is:

\begin{align} \min_{\theta} L(\theta) \text{ or } \max_{\theta} L(\theta)\ \text{subject to:}\ E_j(\theta) = 0, \quad j = 1,2,...,m\ I_k(\theta) \geq 0, \quad k = 1,2,...,n \end{align}

Steps in Problem Formulation

1. Define the objective: Clearly state what needs to be optimized
2. Identify design variables: Determine what can be adjusted
3. Establish a mathematical model: Express the objective as a function of the design variables
4. Identify constraints: Determine limitations on the design variables
5. Verify formulation: Check that the formulation accurately represents the original problem

Examples

Example 1: Beam Design

Objective: Minimize weight of a beam Design variables: Width ($w$) and height ($h$) of the beam cross-section Constraints:

• Maximum stress below yield stress
• Maximum deflection below threshold
• Geometric constraints ($h\ge 2w$, $w\ge 10$ cm)

Mathematical formulation:

• $L(w,h)=\rho Lwh$ (where $\rho$ is density and $L$ is length)
• Subject to stress, deflection, and geometric constraints

Example 2: Portfolio Optimization

Objective: Maximize expected return Design variables: Fraction of capital invested in each asset # Constraints:

• Sum of fractions equals 1
• Maximum risk threshold
• Minimum diversification requirements

Common Pitfalls

1. Incorrect objective function: Failing to capture what truly needs to be optimized
2. Missing constraints: Overlooking important limitations
3. Too many variables: Including unnecessary design variables that complicate the problem
4. Poor scaling: Variables with vastly different magnitudes causing numerical issues
5. Over-constraining: Setting contradictory constraints that leave no feasible solution

Related Pages

• Introduction to Optimisation
• Univariate Optimisation
• Multivariate Optimisation
• Equality Constraints
• Inequality Constraints