Optimization is the process of finding the best solution among the solutions allowed by a problem. It appears in mathematics, engineering, computer science, and economics—for example, when choosing the shortest route, minimizing production costs, or maximizing performance.
An optimization problem asks for the best solution from a set of feasible solutions. A solution is feasible when it satisfies all the requirements and constraints of the problem. The criterion used to compare feasible solutions is called the objective function or, in some settings, the measure or goal function.
All flashcards
What is a feasible solution?
A candidate solution that satisfies every constraint and requirement of the problem.
What is the search space?
The set of candidate points or solutions that meet the problem's requirements and can be evaluated to find an optimum.
What are inequality and equality constraints in a continuous optimization problem?
Inequality constraints have the form gi(x)≤0; equality constraints have the form hj(x)=0. Both restrict which values of x are feasible.
How does a decision problem differ from an optimization problem?
A decision problem asks a yes-or-no question about whether a qualifying solution exists. An optimization problem seeks the best feasible solution or objective value.
What is the purpose of an approximation algorithm?
It aims to produce a near-optimal solution to a difficult optimization problem using a practical amount of computation.
Quiz questions
What does an optimization problem seek?Answer: The best feasible solution under a stated objective.
What does the search space represent in an optimization problem?Answer: The set of candidate solutions that meet the problem's requirements.
Which statement distinguishes continuous optimization from discrete optimization?Answer: Continuous optimization uses continuous-valued variables; discrete optimization selects from countable possibilities.
How can maximizing f(x) over a feasible set be expressed as minimization?Answer: Minimize −f(x) over the same feasible set.
For a graph with vertices u and v, which question is a decision version of finding a shortest path?Answer: Does a path from u to v use 10 or fewer edges?