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Collaborative optimization is examined with a focus on its consistency equality constraints and the effect of their vanishing gradients. Attempts are made to enhance its analytical and computational performance. General theoretical investigation and specific analytical study show that when system-level variables satisfy the consistency equality constraints, system-level optimization generally will encounter no-solution and convergence difficulties if it depends on Lagrange multipliers in the Karush-Kuhn-Tucker necessary conditions. Thus, to avoid such difficulties, algorithms and methods that need no Lagrange multipliers at all are used. Alternatively, to avoid multiplier troubles in applying the Karush--Kuhn--Tucker conditions, the consistency constraints are replaced by other constraints that are not sums of squared differences. Specific computational study shows that enhanced application of the penalty-function method can yield approximate solutions very close to the solution of the predecomposed problem. Convergence is obtained with little difficulty when only the least of the freedom allowable by collaborative optimization is taken in using or producing variants of the targets.