Document Type : Research Article
Authors
Department of Mathematics, Payame Noor University, P.O. Box 19395-4697, Tehran, Iran
10.30473/coam.2026.77019.1395
Abstract
We consider a finite-dimensional multiobjective optimization problem with locally Lipschitz objective functions and finitely many locally Lipschitz inequality constraints. We formulate an active-set Slater test in which σxˆ is selected from a structurally restricted admissible class or selection rule Σ(ψ, xˆ) prescribed before testing the qualification, rather than constructed from an MFCQ direction. Pointwise (σxˆ, rj)-invex inequalities with rj ≥ 0 imply the Clarke–Mangasarian–Fromovitz constraint qualification (ClarkeMFCQ). Standard maximum-function and normal-cone arguments then yield normalized Clarke–KKT multipliers at locally weakly efficient solutions, positive objective multipliers for positive weighted-sum solutions, and a perturbed-KKT inclusion at locally isolated efficient solutions. If Σ(ψ, xˆ) contains every locally Lipschitz map, the test is equivalent to Clarke-MFCQ. A nonsmooth nonconvex example illustrates the assumptions and multiplier inclusions.
Highlights
- An active σ,r-Slater condition is proven sufficient for Clarke-MFCQ.
- The condition is proven equivalent to Clarke-MFCQ over an unrestricted class.
- Fritz–John and normalized KKT conditions are derived for weakly efficient points.
- Positive weighted-sum solutions are shown to admit strictly positive multipliers.
- A regularity-free normal-cone estimate yields a perturbed-KKT condition.
Keywords
- Nonsmooth multiobjective optimization
- Clarke subdifferential
- constraint qualification
- Invexity
- KKT conditions
- Isolated efficient solution
Main Subjects
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