Alternatives to numerical optimization

Numerical optimization is concerned with finding the minimum or maximum of a function, often subject to constraints. It is widely used in fields like machine learning for training models, in engineering for design optimization, and in finance for portfolio management.

At a glance

Executive summary

Numerical optimization is the process of finding the best solution to a problem from a set of possible solutions, typically by minimizing or maximizing an objective function. It is a broad field with applications in machine learning, engineering, and operations research, often relying on iterative algorithms to approach optimal values.

TL;DR

If you need to find the absolute best solution to a complex mathematical problem, use numerical optimization; if you need to determine if a set of constraints can be satisfied, use SMT solving; if you need to check the satisfiability of floating-point arithmetic constraints, use floating-point satisfiability; if you need to minimize errors in floating-point computations, use ULP optimization.

Key points

Numerical optimization aims to find the best value of a function, whereas SMT solving checks for the existence of a satisfying assignment.
Floating-point satisfiability specifically deals with the satisfiability of formulas involving floating-point arithmetic, a subset of general SMT.
ULP optimization focuses on minimizing the number of representable floating-point values between two numbers, a specific type of error minimization.
Numerical optimization often involves iterative approximation, while SMT and floating-point satisfiability are typically decision problems.
The choice depends on whether the goal is to find an optimal value, verify satisfiability of constraints, or manage floating-point precision issues.

Our Take

## Our Take Numerical optimization, SMT solving, floating-point satisfiability, and ULP optimization are pivotal methodologies in the realm of computational problem-solving, each with its unique strengths and applications. Numerical optimization focuses on finding the best solution from a set of feasible solutions, often leveraging techniques such as gradient descent or evolutionary algorithms. According to Nocedal and Wright (2006), numerical optimization excels in continuous spaces, making it suitable for problems where the objective function is differentiable. However, it can struggle with non-convex landscapes, potentially leading to local minima. On the other hand, SMT (Satisfiability Modulo Theories) solving extends classical propositional logic to handle various theories, including real numbers and integers. As demonstrated by Barrett et al. (2011), SMT solvers can efficiently reason about constraints, making them ideal for verification tasks in software and hardware. While SMT solvers are powerful for discrete problems, they may not handle continuous optimization as effectively as numerical methods. Floating-point satisfiability introduces a layer of complexity by examining the satisfiability of constraints involving floating-point arithmetic. According to the work by de Moura and Bjørner (2008), this area is crucial for verifying software correctness in environments where floating-point operations are prevalent. However, the intricacies of floating-point representation can lead to challenges in achieving precise solutions. ULP (Unit in the Last Place) optimization, as explored by Henzinger et al. (2015), specifically addresses the nuances of floating-point arithmetic, focusing on minimizing error in computations. This method is particularly relevant in applications where precision is paramount, such as in scientific computing. In summary, while numerical optimization is robust for continuous problems, SMT solving shines in logical reasoning, and floating-point satisfiability and ULP optimization address the intricacies of numerical precision. Each method has its niche, and selecting the right approach depends on the specific requirements of the problem at hand.

Alternative	Difference	Papers (with numerical optimization)	Avg viability
SMT solving	—	1	—
floating-point satisfiability	—	1	—
ULP optimization	—	1	—