Abstract
Constraint qualification (CQ) is an important concept in nonlinear programming. This paper investigates the motivation of introducing constraint qualifications in developing KKT conditions for solving nonlinear programs and provides a geometric meaning of constraint qualifications. A unified framework of designing constraint qualifications by imposing conditions to equate the so-called ``locally constrained directions' to certain subsets of ``tangent directions' is proposed. Based on the inclusion relations of the cones of tangent directions, attainable directions, feasible directions and interior constrained directions, constraint qualifications are categorized into four levels by their relative strengths. This paper reviews most, if not all, of the commonly seen constraint qualifications in the literature, identifies the categories they belong to, and summarizes the inter-relationship among them. The proposed framework also helps design new constraint qualifications of readers' specific interests.
Highlights
Consider the following general nonlinear programming problem: min f (x) s.t. gi(x) ≤ 0, i ∈ I, hj(x) = 0, j ∈ J, (1)x ∈ Rn, where I and J are index sets, f (x), gi(x), hj(x) are real valued functions on Rn for each i ∈ I and j ∈ J
We study the motivation of introducing constraint qualifications by investigating the development of KKT conditions for problem (1) with J = ∅ and provide a geometric meaning of constraint qualifications
We focus on constraint qualifications in finite dimensional smooth optimization
Summary
Based on the above mentioned, we propose a unified framework of designing constraint qualifications by imposing conditions such that the cone of locally constrained directions equals to certain subsets of the cone of tangent directions. We investigate the motivation of introducing a constraint qualification and its geometric meaning in developing KKT conditions for inequality constrained nonlinear programs. This leads to a unified framework of designing constraint qualifications and a four-level categorization scheme of the constraint qualifications. We can design more constraint qualifications by adding additional assumptions, such as convexity and closeness, on the cones of directions. If not all, of the commonly seen constraint qualifications in the literatures, identify the categories they belong to and show the inter-relations among them
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Journal of Industrial & Management Optimization
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
