Many optimization problems in Matlab are naturally framed as maximization problems, seeking the highest possible value of a function. However, most Matlab optimization toolboxes are designed to work with minimization problems. This means you often need to convert a maximization problem into an equivalent minimization problem before you can solve it using Matlab's powerful functions like fminsearch, fminbnd, or more advanced solvers from the Optimization Toolbox. This guide provides crucial tips to master this conversion.
Understanding the Core Concept: Negation
The fundamental trick to converting a maximization problem into a minimization problem lies in negation. If you want to find the maximum of a function f(x), you can instead find the minimum of -f(x). This is because the point where -f(x) is at its minimum corresponds precisely to the point where f(x) is at its maximum.
Mathematical Proof:
Let's say x* is the point where f(x) achieves its maximum value, i.e., f(x*) ≥ f(x) for all x. Then, multiplying by -1, we get -f(x*) ≤ -f(x) for all x. This shows that -f(x*) is the minimum value of -f(x).
Practical Implementation in Matlab
Let's illustrate this with a simple example. Suppose you have the following function:
f = @(x) x.^2 - 4*x + 5; %A simple parabola
And you want to find the maximum of f(x) within the interval [0, 4]. Instead of directly searching for a maximum, we can define a new function:
g = @(x) -f(x); %The negation of f(x)
Now, we can use a Matlab minimization function, such as fminbnd, to find the minimum of g(x):
[xmin, fmin] = fminbnd(g, 0, 4);
xmax = xmin; %The x-value of the minimum of g(x) is also the x-value of the maximum of f(x)
fmax = -fmin; %The minimum value of g(x) is the negative of the maximum of f(x)
This code will find the xmin that minimizes g(x), which is the same xmax that maximizes f(x). fmax will be the maximum value of f(x).
Handling Constraints
The conversion process remains the same even when constraints are involved. If your maximization problem includes constraints, simply apply the negation to the objective function and keep the constraints unchanged. For example, if your original problem is:
Maximize f(x) subject to g(x) ≤ 0 and h(x) = 0
The equivalent minimization problem becomes:
Minimize -f(x) subject to g(x) ≤ 0 and h(x) = 0
Advanced Optimization Techniques
For more complex problems, Matlab's Optimization Toolbox offers a range of powerful solvers that handle various types of constraints and objective functions. Remember to always negate the objective function when working with maximization problems using these solvers. Consult the Matlab documentation for specific guidance on using each solver.
Beyond Negation: Alternative Approaches (Less Common)
While negation is the most straightforward approach, alternative methods exist for specific problem structures. These are less commonly used but worth mentioning:
- Transformation: In some cases, a mathematical transformation might simplify the problem, making it easier to work with directly as a minimization. This requires a deep understanding of the problem's underlying mathematical structure.
- Reformulation: Sometimes, the problem can be re-framed to naturally lead to a minimization problem without needing to explicitly negate the objective function.
Conclusion
Converting maximization problems to minimization problems in Matlab is a fundamental skill for any user of its optimization toolboxes. By mastering the simple technique of negation and understanding its implications, you can unlock the full power of Matlab's optimization capabilities to solve a wide range of problems. Remember to always check your results and ensure they make intuitive sense within the context of your problem.
