Overview: Applications Optimization
General
On the following subpages, you will find information about the types of problems that can be solved using mathematical optimization. The methods used and the basic theory are briefly outlined; code examples detail practical applications.
Relevance
Whenever it comes to choosing between multiple options, the question of the best option naturally arises — an optimization problem. The specific application may involve choosing product parameters, designing schedules, estimating unknown quantities, or making decisions in complex systems. Hence, optimization is now found everywhere and is used in finance, estimation theory, optimal control, scheduling, industrial product design, metrology, telecommunications, and much more.
Mathematical optimization deals with the minimization or maximization of an objective function \(f(x)\) under constraints for the variable \(x\). The formal notation for this task is
$$ \begin{align} \min_x ~~~&f(x) \\ \text{subject to} ~~~&x \in D \end{align}$$
This encodes the instructions that an \(x\) is to be found which minimizes the function value \(f(x)\) (\(\min _x f(x)\)). Here, \(x\) can only be chosen from a specific set \(D\) (\(x\in D\)), which formalizes the requirements for the optimization variable \(x\).
Examples
The problem formulated above includes a large set of tasks relevant to practice. Often, \(f(x)\) quantifies costs, material expenditure, durations, uncertainties, probabilities of error, or other negative performance measures whose reduction is desired. The (often high-dimensional) variable \(x\) represents decisions with direct impacts on the performance measures; such as investment decisions, product properties, scheduling, experimental designs, or safety margins. These are subject to user-defined demands that, for example, formalize the adherence to certain standards. The following illustration demonstrates this, and the table lists exemplary interpretations.
| Example | \(f(x)\) | \(x_1,x_2, …\) | \(D\) |
|---|---|---|---|
| Stock portfolio | - Expected profit | Investment into stock 1, 2, … | Acquisition constraints |
| Truss topology design | Material consumption | Length, thickness of bars | Stability certificate |
| Facility layout | Production times | Location devices 1, 2, … | Machine neighborhoods |
| Pharmaceutical tests | Uncertainty, significance level | Tests of class 1,2, … | Depencies of test classes |
| Transportation planning | Time effort | Sequence of locations | Transportation capacity |
| Job scheduling | Workload | Starting times of tasks | Job sequences |
Types of tasks
Many practical tasks from disciplines such as finance, business administration, logistics, physics, engineering, telecommunications, game theory, data analysis, … can be formulated as the optimization problem \(\min _x f(x), x\in D\). Regardless of the specific application, we identify three classes of tasks into which tasks from the above disciplines can be categorized:
Optimal design
The optimization variables \(x\) represent static design parameters of a product. They are to be chosen, taking into account a model based on preliminary considerations, such that the product maximizes a certain performance measure \(-f(x)\) while ensuring certain properties are maintained.
This class of tasks includes, for example, experimental design, portfolio optimization, transport, routing, and logistics problems, as well as the optimization of machines and structures and their arrangement.
Optimal estimation
The optimization variable \(x\) represents parameters of a predictive model \(g(x,z)\) with independent variable \(z\). The model aims to make predictions about which outcome \(l\) occurs, depending on a previously made observation \(z\).
The relationship between \(z\) and \(l\) is to be modeled, where datasets \(\{z_k,l_k\}_{k=1}^n\) have been observed, and the model \(g(x,z)\) needs to be adjusted so that it contradicts the data foundation as little as possible. This class of tasks includes classic regression problems of parameter estimation and reverse engineering, but also spatial statistics, prediction, learning of correlation structures, statistical signal separation, and reconstruction. These tasks occur, for example, in complex measurement problems.
Optimal control
The optimization variable \(x\) represents a sequence of control inputs in a system that behaves dynamically and possibly randomly. This input must be chosen so that the discrepancy \(f(x)\) between the development carried out by the system and a preferred target development is minimized.
In this process, the control variables \(x\) may be limited to a specific area \(D\). Problems of this task class include sequential statistical decision problems such as traffic flow regulation, supply chain management, pricing for financial derivatives, adaptive medical treatment, as well as tasks of classical control engineering.
Outlook
Tasks from all the aforementioned areas have already been successfully formulated and solved as mathematical optimization problems. The following selection includes simplified example problems that are non-trivial to solve without the appropriate tools. Illustrations and brief descriptions showcase the problem, and a commented formulation as a mathematical optimization problem clarifies the connections between real-world issues and the mathematical modeling level. We use special formal arrangements of objective functions and equations and inequalities that can be reliably solved through linear programming, quadratic programming, semidefinite programming, and dynamic programming.
