The paper formulates a rotated-quadratic cone (Rcone) technique for producing a variable allpass-delay filter (AP-DF) while significantly containing the delay-error overshoots. After showing that both frequency-response (FR) error and delay-error have denominators and numerators being linear functions of the AP-DF parameters, the paper reveals a minimization scheme for designing a variable AP-DF. The design technique uses the Rcone-programming to minimize the FR-error peak while effectively containing the delay-error overshoots. Since the Rcone-programming design reduces to a convex minimization, performing the convex minimization yields a convergent AP-DF solution. The paper provides an AP-DF design example to exemplify the utility of the Rcone-programming formulation. As compared to other design techniques, this new scheme can reduce the design to a convex-programming problem, rendering it easy to solve. The design simulations also verify that this Rcone-programming formulation can further suppress the delay overshoots as compared with other design techniques. More precisely, the Rcone-programming design causes the smallest delay-error peak (0.002960) among all the designs.