How do you test the accuracy of nonlinear programming solutions?

Nonlinear programming (NLP) is a branch of operations research that deals with optimizing functions that are not linear, such as quadratic, exponential, or logarithmic. NLP can be used to model and solve complex problems in engineering, economics, finance, and other fields. However, NLP solutions are not always easy to verify, as they may depend on the choice of algorithm, initial point, tolerance, and other factors. How do you test the accuracy of nonlinear programming solutions? In this article, you will learn some methods and tips to check your NLP results and avoid common pitfalls.

1 Compare with linearization

One way to test the accuracy of NLP solutions is to compare them with the solutions obtained by linearizing the nonlinear function. Linearization is a process of approximating a nonlinear function by a linear one, usually by using the first-order Taylor expansion. Linearization can simplify the problem and make it easier to solve, but it may also introduce errors and lose some information. Therefore, you can use linearization as a benchmark to see how close your NLP solution is to the linear one, and how much improvement you can get from using NLP. However, you should also be aware of the limitations of linearization, such as the validity of the approximation, the sensitivity to the expansion point, and the possible loss of convexity or feasibility.

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Frederik Geth
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Convex relaxation through reformulation and lifting is equally powerful as linearization , it establishes a lower bound on the true objective value.

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Sebastián Eduardo Arpón Barba

PhD in Management, expert in optimization and machine learning scientist
(edited)
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If the problem is convex as those mentioned in the intro it could aproximated by piecewise linear aproximations, which depending on the number of pieces could lead to good aproximation, but now the problem is a mixed integer linear problem.

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Pruthiraj Swain, SMIEEE

Microgrid | BESS | DERMS | AI/ML | Forecasting | Optimization | Fuel Cell | Hydrogen Ecosystem | Design Thinking Facilitator | SAFe® Practitioner | SAFe® POPM | PSM®-I | LSSGB | MHFAider®
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Instead of using a fixed linearization approach for all NLP solutions, dynamically adjust the threshold for linearization based on the complexity of the function being analyzed. More complex functions could have stricter linearization criteria, while simpler functions might allow for a looser linearization approach. This adaptive thresholding could lead to more accurate benchmarking and better insights into the NLP solution's performance.

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Xiaowei Bao

Data Science/Decision Science Leader at Hyatt
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In most situations, using just one linearization doesn't provide accurate estimates. Convex relaxation is more effective but can be tougher to solve. In my experience, combining several linear relaxations together usually works better.

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2 Use multiple algorithms

Another way to test the accuracy of NLP solutions is to use multiple algorithms and compare their results. There are many algorithms for solving NLP problems, such as gradient-based methods, Newton-type methods, trust-region methods, penalty methods, interior-point methods, and global optimization methods. Each algorithm has its own advantages and disadvantages, such as speed, robustness, convergence, and scalability. By using different algorithms, you can see how consistent your NLP solution is across different methods, and how sensitive it is to the choice of algorithm. You can also use multiple algorithms to find the best solution among them, or to combine their strengths and weaknesses.

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Pruthiraj Swain, SMIEEE

Microgrid | BESS | DERMS | AI/ML | Forecasting | Optimization | Fuel Cell | Hydrogen Ecosystem | Design Thinking Facilitator | SAFe® Practitioner | SAFe® POPM | PSM®-I | LSSGB | MHFAider®
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This test can be done by keeping in mind following aspects: - Convergence rates - Computational time - Resilience to external and internal disturbances - Optimal range of hyper-parameters for different applications

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3 Check the optimality conditions

A third way to test the accuracy of NLP solutions is to check the optimality conditions of the problem. Optimality conditions are mathematical criteria that must be satisfied by a solution to be optimal, such as the Karush-Kuhn-Tucker (KKT) conditions for constrained NLP problems. Optimality conditions can help you verify whether your NLP solution is indeed optimal, or whether it is only a local optimum, a feasible point, or an infeasible point. You can check the optimality conditions by calculating the gradient, the Lagrange multiplier, the complementary slackness, and the feasibility of your NLP solution, and comparing them with the required values. You can also use the optimality conditions to diagnose and correct any errors or violations in your NLP solution.

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Lior Prechner Davidovitch

Senior Data Scientist @ Pagaya | PhD in Decision Theory
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I think this whole essay doesn't distinguish between two situations: - When the function is smooth and convex, we are likely to end up in the vicinity of the optimum, and the question is are we close enough? - When the function is not, we are likely to end up in a local optimum. In this case, the question can be are the results adequate? Or, are we close enough? The techniques in both cases may be different.

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4 Evaluate the objective function

A fourth way to test the accuracy of NLP solutions is to evaluate the objective function of the problem at the solution point. The objective function is the function that you want to minimize or maximize in NLP problems, such as the cost, profit, or utility. Evaluating the objective function can help you measure the quality and performance of your NLP solution, and see how much it differs from the optimal value. You can also evaluate the objective function at different points around the solution point, and see how it changes with respect to the variables. This can help you assess the sensitivity, stability, and robustness of your NLP solution, and identify any potential errors or improvements.

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Lior Prechner Davidovitch

Senior Data Scientist @ Pagaya | PhD in Decision Theory
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Evaluating the objective function can be used as a sanity check. Other than that, it's an answer to a different question - is the NLP result adequate? Remember that non-optimal solutions can also do the job, sometimes... Checking the value around the point is similar to estimating the gradient.

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5 Perform sensitivity analysis

A fifth way to test the accuracy of NLP solutions is to perform sensitivity analysis of the problem. Sensitivity analysis is a technique of studying how the NLP solution changes with respect to the parameters, constraints, or data of the problem. Sensitivity analysis can help you understand the impact and influence of different factors on your NLP solution, and how they affect its optimality, feasibility, and reliability. You can perform sensitivity analysis by varying the values of the parameters, constraints, or data within a certain range, and observing how the NLP solution responds. You can also use sensitivity analysis to test the validity and accuracy of your assumptions, models, and data, and to explore alternative scenarios and outcomes.

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6 Use simulation and validation

A sixth way to test the accuracy of NLP solutions is to use simulation and validation of the problem. Simulation and validation are methods of testing and verifying your NLP solution by using real or synthetic data, experiments, or cases. Simulation and validation can help you evaluate the applicability and effectiveness of your NLP solution in practice, and see how it performs under different conditions and situations. You can use simulation and validation by applying your NLP solution to the actual or simulated data, experiments, or cases, and comparing the results with the expected or observed values. You can also use simulation and validation to identify and correct any discrepancies, errors, or limitations in your NLP solution.

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7 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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Pruthiraj Swain, SMIEEE

Microgrid | BESS | DERMS | AI/ML | Forecasting | Optimization | Fuel Cell | Hydrogen Ecosystem | Design Thinking Facilitator | SAFe® Practitioner | SAFe® POPM | PSM®-I | LSSGB | MHFAider®
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02 major ways which are missed out are Scalability testing and Solution stability with minimal variations in the initial conditions.

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How do you test the accuracy of nonlinear programming solutions?

1

2

3

4

5

6

7

1 Compare with linearization

2 Use multiple algorithms

3 Check the optimality conditions

4 Evaluate the objective function

5 Perform sensitivity analysis

6 Use simulation and validation

7 Here’s what else to consider

Operations Research

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How do you test the accuracy of nonlinear programming solutions?

1

2

3

4

5

6

7

1 Compare with linearization

2 Use multiple algorithms

3 Check the optimality conditions

4 Evaluate the objective function

5 Perform sensitivity analysis

6 Use simulation and validation

7 Here’s what else to consider

Operations Research

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