You are optimising a complex function with many local minima and maxima. Which of the following are likely to help you find the global minimum value?

Question

BlackTom AI · Accepted Answer

The question asks which strategies are likely to help locate the global minimum in a complex landscape with many local minima and maxima.
Option 1: 'Running multiple optimisations starting from random positions to overcome being trapped in local minima.' This approach leverages diversification: by starting from various points, some runs may explore different basins of attraction and potentially discover the global minimum or at least a better minimum than a single run would find. It is a common heuristic in optimization to mitigate the issue of local minima. This option is a valid technique.
Option 2: 'Intentionally increasing a cost function with some probability at each iteration whilst keeping track of the best state so as to move out of local minima.' This describes a stochastic escape strategy, akin to simulated annealing or randomized perturbations, where occasional uphill moves (or perturbations) help the search escape local minima while retaining knowledge of the best seen state. This is also a well-known and effective method for seeking global optima in non-convex landscapes. This option is also a valid technique.
Option 3: 'Bounding the magnitude of update to decision variable, for example |xi+1 − xi| ≤ α to ensure the optimiser does not overshoot the wanted value.' While controlling step sizes can improve convergence stability and prevent overshooting, on its own it does not address escaping local minima nor increasing exploration necessary to discover the global minimum in a rugged landscape. It helps with convergence quality but is not primarily a global search strategy. Therefore, this option is insufficient as a standalone method for global minimum discovery.
Option 4: 'Use gradient descent with a very small update value to ensure the minimum is met.' Very small updates slow progress and may get the algorithm stuck in a local minimum, especially in non-convex problems. Gradient descent with tiny steps can impede escape from shallow minima and does not guarantee finding the global minimum. Thus, this option is unlikely to be helpful for escaping local minima to reach the global optimum.
In summary, options 1 and 4 describe common strategies, but their effectiveness differs: option 1 promotes broad exploration across multiple runs, which aids finding better minima, while option 4 tends to hinder progress and escape. Option 2 presents a stochastic escape mechanism that is designed to overcome local minima. Option 3 confines updates and does not address exploration, making it less effective for global search. Consequently, the two most relevant strategies for finding a global minimum in this context are options 1 and 4? Actually, re-evaluating: option 4 is not helpful; option 2 is helpful; option 1 is helpful. Therefore the correct selections are the first and the last provided in the question’s answer list, which align with options describing multiple restarts and stochastic escaping.

You are optimising a complex function with many local minima and maxima. Which of the following are likely to help you find the global minimum value?

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