Constrained Minimization

Author

Affiliation

Enrico Bertolazzi

University of Trento, Department of Industrial Engineering

This part develops constrained minimization: how to recognize, derive and verify a minimum of a function subject to equality and inequality constraints. The thread runs from the method of Lagrange multipliers, which handles equalities, to the Karush-Kuhn-Tucker (KKT) conditions, which extend it to inequalities, and closes with the numerical methods that solve the resulting systems.

TipPrerequisites, objectives, and expected outcomes

Prerequisites: multivariable calculus, Taylor expansion, and elementary linear algebra.

Objectives: formulate first- and second-order optimality conditions, interpret constraint qualifications, and connect KKT systems with numerical algorithms.

Expected outcomes: the reader can derive and verify KKT conditions, classify candidate points, diagnose degeneracy, and select an unconstrained or constrained numerical method.

Contents

• Mathematical Preliminaries
• Lagrange Multipliers
• Sylvester’s Criterion
• Inequality Constraints and KKT Conditions
• KKT Worked Examples (companion volume)
• Optimization Algorithms
  • Unconstrained Optimization
  • Constrained Optimization

[1]

Dimitri P. Bertsekas. 2016. Nonlinear programming (3rd ed.). Athena Scientific.

[2]

Roger Fletcher. 2000. Practical methods of optimization (2nd ed.). John Wiley & Sons.

[3]

Jorge Nocedal and Stephen J. Wright. 2006. Numerical optimization (2nd ed.). Springer Science & Business Media. https://doi.org/10.1007/978-0-387-40065-5