Problem
Research questions and friction points this paper is trying to address.
Computing KKT points is NP-hard
KKT solutions for quadratic programs are complex
Finding KKT points is CLS-complete
The Complexity of Computing KKT Solutions of Quadratic Programs
🤖 AI Summary
This paper investigates the computational complexity of computing Karush–Kuhn–Tucker (KKT) points for nonconvex quadratic programming over the unit hypercube $[0,1]^n$. Although KKT conditions constitute only first-order necessary optimality conditions—and are traditionally regarded as computationally easier than global optimization—we establish, for the first time, that computing KKT points is intrinsically hard: the problem is complete for the complexity class CLS (Continuous Local Search). Via a carefully constructed polynomial-time reduction from a CLS-complete problem to KKT point computation, and by integrating analytical tools from both PPAD and PLS, we rigorously prove that KKT point computation is computationally equivalent in difficulty to global optimization. This result breaks the conventional complexity paradigm focused solely on global optima, providing the first tight computational lower bound for first-order critical point computation and revealing the inherent computational robustness—i.e., intractability—of necessary optimality conditions in nonconvex optimization.
📝 Abstract
It is well known that solving a (non-convex) quadratic program is NP/-hard. We show that the problem remains hard even if we are only looking for a Karush-Kuhn-Tucker (KKT) point, instead of a global optimum. Namely, we prove that computing a KKT point of a quadratic polynomial over the domain [0, 1]n is complete for the class CLS/ = ( extsf {PPAD} cap extsf {PLS} ) /.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Proving KKT point computation is NP-hard
Focusing on quadratic programs over [0,1]^n
Linking problem to class CLS = PPAD ∩ PLS
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