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What are some of the common pitfalls and errors to avoid in numerical optimization in finance?

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Numerical optimization is a powerful technique for finding the optimal values of variables that maximize or minimize a given objective function, such as profit, risk, or utility. However, numerical optimization in finance also involves some challenges and pitfalls that can lead to inaccurate or unreliable results. In this article, you will learn about some of the common sources of errors and how to avoid them in your numerical optimization projects.

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Toni Esteves

1 Local vs global optima

One of the main difficulties in numerical optimization is finding the global optimum, which is the best possible solution among all feasible ones, rather than a local optimum, which is only the best solution in a small neighborhood of the search space. Local optima can be misleading and trap the optimization algorithm in suboptimal regions, especially if the objective function is nonlinear, multimodal, or noisy. To avoid this pitfall, you should use a global optimization method, such as genetic algorithms, simulated annealing, or particle swarm optimization, that can explore the search space more widely and escape from local optima.

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Toni Esteves
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In my expertise dealing with the challenges posed by global and local optima in optimization problems requires careful consideration and the application of suitable strategies. In that way, we can apply Random Initialisation, Population-based Methods, or Restart Strategies. There are also some strategies based on Exploration and exploration like Adaptive Strategies and Multi-armed Bandit Techniques. Related to Problem-specific Strategies we can apply Problem Decomposition and Constraint Handling.

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2 Convergence and termination criteria

Another challenge in numerical optimization is determining when to stop the optimization process and accept the current solution as the optimal one. This is usually done by setting some convergence or termination criteria, such as the maximum number of iterations, the maximum running time, or the minimum improvement in the objective function value. However, these criteria can be tricky to choose and may depend on the problem size, complexity, and structure. If the criteria are too loose, the optimization may end prematurely and miss better solutions. If the criteria are too tight, the optimization may take too long and waste computational resources. To avoid this pitfall, you should test different criteria and compare the results for robustness and efficiency.

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Toni Esteves
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Dealing with convergence and termination criteria is crucial for ensuring the effectiveness and efficiency of optimization algorithms. Some examples of how to deal with this are whether the objective function value has stabilized or reached a plateau or the change in the objective function value or solution parameters falls below a predefined threshold. We can define the maximum number of iterations or evaluations the algorithm has executed or if the algorithm has exceeded a specified time limit.

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3 Constraints and feasibility

Many optimization problems in finance involve constraints that limit the range or values of the variables, such as budget, risk, or regulatory restrictions. These constraints can make the optimization problem more difficult and complex, as they reduce the feasible region of the search space and may introduce nonlinearities or discontinuities in the objective function. Moreover, some optimization methods may not handle constraints well and may generate infeasible solutions that violate the constraints. To avoid this pitfall, you should use a constraint-handling technique, such as penalty functions, barrier functions, or projection methods, that can enforce the constraints and penalize or repair infeasible solutions.

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Toni Esteves
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Despite the mentioned strategy of using Constraint-handling techniques, we can incorporate Constraints into the Optimisation Process providing relaxation. In that way, we modify the objective function to penalize infeasible solutions or reward feasibility and add constraint functions to the objective function to explicitly enforce constraints. We can explore feasible regions by designing the optimization algorithm to prioritize the exploration of feasible regions of the search space while avoiding or penalizing infeasible regions.

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4 Sensitivity and stability

Another issue in numerical optimization is how sensitive and stable the optimal solution is to changes in the input data, parameters, or assumptions. This is important to assess the reliability and robustness of the optimization results and to account for uncertainty and variability in the financial environment. However, some optimization methods may not provide sufficient information or measures of sensitivity and stability, such as confidence intervals, error bounds, or perturbation analysis. To avoid this pitfall, you should use a sensitivity analysis technique, such as Monte Carlo simulation, scenario analysis, or bootstrap methods, that can evaluate the impact of different sources of uncertainty and variability on the optimal solution.

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5 Implementation and validation

The final challenge in numerical optimization is how to implement and validate the optimization results in practice. This involves translating the mathematical formulation of the optimization problem into a computer code that can run efficiently and accurately on a given platform or software. It also involves checking the validity and quality of the optimization results by testing them against historical data, benchmarks, or other criteria. However, some optimization methods may not be easy to implement or validate, as they may require complex algorithms, specialized libraries, or proprietary software. To avoid this pitfall, you should use a standard and reliable optimization tool, such as MATLAB, Python, or Excel, that can handle different types of optimization problems and methods and provide useful features and functions for implementation and validation.

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6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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What are some of the common pitfalls and errors to avoid in numerical optimization in finance?

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1 Local vs global optima

2 Convergence and termination criteria

3 Constraints and feasibility

4 Sensitivity and stability

5 Implementation and validation

6 Here’s what else to consider

Numerical Analysis

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