# American Institute of Mathematical Sciences

October  2013, 9(4): 983-1001. doi: 10.3934/jimo.2013.9.983

## On constraint qualifications: Motivation, design and inter-relations

 1 Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC, 27695, United States 2 Department of Industrial and System Engineering, North Carolina State University, Raleigh, NC 27695 3 Department of Mathematical Sciences, Tsinghua University, Beijing, 100084

Received  February 2013 Revised  March 2013 Published  August 2013

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.
Citation: Ziteng Wang, Shu-Cherng Fang, Wenxun Xing. On constraint qualifications: Motivation, design and inter-relations. Journal of Industrial & Management Optimization, 2013, 9 (4) : 983-1001. doi: 10.3934/jimo.2013.9.983
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show all references

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