Classical computing still suffers from the high computational complexity of finding the extreme value of a constrained goal function or an unordered database with respect to a certain constraint. Quantum computing offers new possibilities and solutions due to its quantum nature to surmount this problem. In this paper, we extended a new version of quantum existence testing (QET) _we call it a constrained quantum relation testing (CQRT)_ we also investigate analytically its classical and quantum certainty. Furthermore, we show how the CQRT can be applied for extending the quantum extreme value searching algorithm (QEVSA) version to a constrained quantum optimization algorithm (CQOA). And we gave an extensive explanation of how the CQOA works and how it finds the extreme constrained optimum value solution with high complexity reduction, maximum exponential speed, and high certainty.