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This work introduces a Partition Bound Particle Swarm Optimization (PB-PSO) algorithm to enhance convergence rates in analog circuit optimization. Two new parameters, <inline-formula> <tex-math notation="LaTeX">$\zeta _{1}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\zeta _{2}$ </tex-math></inline-formula>, are incorporated to adaptively update particle velocities based on iteration numbers. The parameter <inline-formula> <tex-math notation="LaTeX">$\zeta _{1}$ </tex-math></inline-formula> depends on the non-linear convergence factor (<inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>) and the number of iterations, <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula>. The results indicate that <inline-formula> <tex-math notation="LaTeX">$\zeta _{1}$ </tex-math></inline-formula>&#x2019;s optimal value occurs with <inline-formula> <tex-math notation="LaTeX">$\alpha = 4.~\zeta _{2}$ </tex-math></inline-formula> partitions iterations into two regions, aiding local and global search. The PB-PSO algorithm, implemented in Python, demonstrates higher convergence rates than existing methods, with successful designs verified through Cadence-Virtuoso circuit simulations. The proposed PB-PSO algorithm converges in 15 and 13 iterations for differential amplifier and two-stage op-amp respectively. For a case study of two-stage amplifier, it achieves a gain of 60.4 dB with a phase margin of 79.76&#x00B0;, meeting input specifications within constraints. The figure of merit was then evaluated using the obtained parameters, which turns out to be 0.275 <inline-formula> <tex-math notation="LaTeX">$V^{-2}$ </tex-math></inline-formula>.