Active-set method

Definition

The active-set method is an optimization technique used to solve constrained optimization problems, particularly in quadratic programming. It focuses on identifying the set of constraints that are active at the solution point, which means they are tight and can influence the optimal solution. By systematically adjusting this set of active constraints, the method iteratively refines the solution until it converges to an optimal point.

5 Must Know Facts For Your Next Test

1. The active-set method begins with an initial guess for the active constraints and refines this set as iterations progress.
2. This method can efficiently handle both equality and inequality constraints by updating the active set accordingly.
3. It uses concepts from linear programming and can leverage solutions from simpler optimization problems to aid in finding solutions to more complex ones.
4. The method is particularly well-suited for problems with a small number of active constraints compared to the total number of variables.
5. Convergence to the optimal solution is guaranteed under certain conditions, making it a reliable choice for quadratic programming problems.

Review Questions

• How does the active-set method determine which constraints are active during the optimization process?
  • The active-set method identifies active constraints by evaluating which constraints are binding at a given solution point. Initially, a set of constraints is selected based on the starting solution, and as the optimization progresses, it assesses if these constraints still influence the objective function. Constraints that remain tight at the optimal solution are retained in the active set, while those that do not contribute may be dropped, allowing for a refined search for the optimum.
• Discuss how the active-set method can be applied in quadratic programming problems with both equality and inequality constraints.
  • In quadratic programming, the active-set method applies by first identifying which inequality constraints are binding at the current solution. For equality constraints, they are always considered active. The algorithm then iteratively updates the set of active constraints based on whether they continue to hold at subsequent iterations. By dynamically managing both sets of constraints, it effectively navigates towards a feasible region that leads to optimal solutions while ensuring all necessary conditions are met.
• Evaluate the efficiency of the active-set method compared to other optimization techniques in solving constrained problems.
  • The efficiency of the active-set method lies in its ability to focus on a smaller subset of relevant constraints during each iteration, which can significantly reduce computational complexity compared to methods that evaluate all constraints simultaneously. While other methods like interior-point algorithms may excel in large-scale problems, active-set methods tend to perform better when there are fewer active constraints relative to total variables. This targeted approach often leads to faster convergence rates and more efficient utilization of computational resources in quadratic programming applications.

Related terms

Quadratic Programming:

A type of mathematical optimization problem where the objective function is quadratic and the constraints are linear.

Constraint: A restriction or condition that the solution to an optimization problem must satisfy.

Lagrange Multiplier:

A technique used to find the local maxima and minima of a function subject to equality constraints, often utilized in conjunction with the active-set method.