# Combinatorial optimization

Within the field of mathematical optimization, combinatorial optimization represents a sub topic with several techniques for finding the optimal solution from a finite (and huge) set of discrete candidate solutions. Combinatorial optimization is the key competence area for the Optimization Group at SINTEF Digital. The main methods we use to solve these optimization problems are listed below.

## Heuristics and metaheuristics

In computer science and mathematical optimization, a metaheuristic is a higher-level procedure or heuristic designed to find, generate, or select a heuristic (partial search algorithm) that may provide a sufficiently good solution to an optimization problem, especially with incomplete or imperfect information or limited computation capacity.

## Constraint programming

In computer science, constraint programming is a programming paradigm wherein relations between variables are stated in the form of constraints. Constraints differ from the common primitives of imperative programming languages in that they do not specify a step or sequence of steps to execute, but rather the properties of a solution to be found.

## Linear programming

Linear programming (LP) (also called linear optimization) is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (mathematical optimization).

## Mixed-integer programming

Mixed integer linear programming (MILP) involves problems in which only some of the variables, xi are constrained to be integers, while other variables are allowed to be non-integers.

## Dynamic programming

In mathematics, management science, economics, computer science, and bioinformatics, dynamic programming (also known as dynamic optimization) is a method for solving a complex problem by breaking it down into a collection of simpler subproblems, solving each of those subproblems just once, and storing their solutions – ideally, using a memory-based data structure.

## Real-time optimisation

Online optimization is a field of optimization theory, more popular in computer science and operations research, that deals with the optimization problems having incomplete knowledge of the future (online).

## Distributed optimisation

Distributed optimization is a problem in which a group of agents must distributedly choose values for a set of variables such that the cost of a set of constraints over the variables is either minimized or maximized.

## Multi-objective optimisation

Multi-objective optimization (also known as Pareto optimization) is an area of multiple criteria decision making, that is concerned with mathematical optimization problems involving more than one objective function to be optimized simultaneously. Multi-objective optimization has been applied in many fields of science, including engineering, economics and logistics (see the section on applications for detailed examples) where optimal decisions need to be taken in the presence of trade-offs between two or more conflicting objectives. Minimizing cost while maximizing comfort while buying a car, and maximizing performance whilst minimizing fuel consumption and emission of pollutants of a vehicle are examples of multi-objective optimization problems involving two and three objectives, respectively. In practical problems, there can be more than three objectives.

## Robust optimisation

Robust optimization is a field of optimization theory that deals with optimization problems in which a certain measure of robustness is sought against uncertainty that can be represented as deterministic variability in the value of the parameters of the problem itself and/or its solution.

## Stochastic programming

In the field of mathematical optimization, stochastic programming is a framework for modeling optimization problems that involve uncertainty. Whereas deterministic optimization problems are formulated with known parameters, real world problems almost invariably include some unknown parameters.

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