Optimal decision-analytic designs are deterministic. Such designs are appropriately criticized in the context of clinical trials because they are subject to assignment bias. On the other hand, balanced randomized designs may assign an excessive number of patients to a treatment arm that is performing relatively poorly. We propose a compromise between these two extremes, one that achieves some of the good characteristics of both. We introduce a constrained optimal adaptive design for a fully sequential randomized clinical trial with k arms and n patients. An r-design is one for which, at each allocation, each arm has probability at least r of being chosen, 0 ⩽ r ⩽ 1/k. An optimal design among all r-designs is called r-optimal. An r1-design is also an r2-design if r1 ⩾ r2. A design without constraint is the special case r = 0 and a balanced randomized design is the special case r = 1/k. The optimization criterion is to maximize the expected overall utility in a Bayesian decision-analytic approach, where utility is the sum over the utilities for individual patients over a ‘patient horizon’ N. We prove analytically that there exists an r-optimal design such that each patient is assigned to a particular one of the arms with probability 1 − (k − 1)r, and to the remaining arms with probability r. We also show that the balanced design is asymptotically r-optimal for any given r, 0 ⩽ r < 1/k, as N/n → ∞. This implies that every r-optimal design is asymptotically optimal without constraint. Numerical computations using backward induction for k = 2 arms show that, in general, this asymptotic optimality feature for r-optimal designs can be accomplished with moderate trial size n if the patient horizon N is large relative to n. We also show that, in a trial with an r-optimal design, r < 1/2, fewer patients are assigned to an inferior arm than when following a balanced design, even for r-optimal designs having the same statistical power as a balanced design. We discuss extensions to various clinical trial settings.