Objective space is a multidimensional space where each dimension represents a different objective or criterion that needs to be optimized in a multi-objective optimization problem. The goal is to identify solutions that offer the best possible trade-offs among competing objectives, often represented as a Pareto front within this space.
The Simplex Algorithm is a popular method for solving linear programming problems by iteratively moving along the edges of the feasible region to find the optimal vertex. It efficiently navigates through feasible solutions in a systematic way, making it a cornerstone technique in operations research and optimization.
An interior point is a point that lies within the boundary of a set in a topological space, meaning there exists a neighborhood entirely contained within the set. This concept is fundamental in topology and optimization, where it is used to determine feasible regions and solutions within constraints.
Lagrangian Multipliers are a mathematical tool used in optimization to find the local maxima and minima of a function subject to equality constraints. By introducing auxiliary variables (the multipliers), this method transforms a constrained problem into an unconstrained one, allowing for easier solution derivation using partial derivatives.
Interior Point Methods are a class of algorithms used to solve linear and nonlinear convex optimization problems by traversing the interior of the feasible region. They are known for their polynomial-time complexity and efficiency in handling large-scale problems compared to traditional simplex methods.
A non-negative function is a mathematical function that only outputs zero or positive values, regardless of its input. This characteristic is crucial in various fields such as probability, where it ensures that probabilities are always non-negative, and in optimization, where it helps in defining feasible regions.
The central path is a trajectory in optimization that solutions of interior-point methods follow as they progress toward an optimal point in a convex optimization problem. It serves as a crucial guide for navigating feasible regions while ensuring convergence to the global optimum efficiently.
Primal feasibility refers to a condition in linear programming where all constraints of the primal problem are satisfied without violating any specified limits. Ensuring primal feasibility is crucial for attaining an optimal solution to the problem, which necessitates adherence to both equality and inequality constraints defined within the system.