A breakpoint search approach for convex resource allocation problems with bounded variables

Abstract

We present an efficient approach to solve resource allocation problems with a single resource, a convex separable objective function, a convex separable resource-usage constraint, and variables that are bounded below and above. Through a combination of function evaluations and median searches, information on whether or not the upper-and lowerbounds are binding is obtained. Once this information is available for all upper and lower bounds, it remains to determine the optimum of a smaller problem with unbounded variables. This can be done through a multiplier search procedure. The information gathered allows for alternative approaches for the multiplier search which can reduce the complexity of this procedure.

Algorithms for the continuous nonlinear resource allocation problem—New implementations and numerical studies

“…Further when we determine if x j ∈ M we only need to check if this holds for j ∈ U + or j ∈ L − depending on if we update the lower or upper bound of the dual variable. This differs from the algorithm in [DWW12] since that algorithm finds M from U + ∪ L − . Note that we make use of information from earlier iterations when updating δ.…”

“…For 5-sets pegging we also determine if a variable is larger than the lower bound or smaller than the upper bound. 5-sets pegging for problem (1) is used in [DWW12], generalizing a method from [Kiw08a]. Assuming that we know that x * j < u j , there is no need to check if x k j equals the upper bound; this might reduce future calculations.…”

“…Assuming that we know that x * j < u j , there is no need to check if x k j equals the upper bound; this might reduce future calculations. The proof of the following proposition follows from the monotonicity of g j and x j (µ) (see [DWW12] for a proof).…”

“…If we let the initial values of the limits be µ = −∞ and µ = ∞, then µ, µ / ∈ [µ l j , µ u j ] and we note that there is no need to check if j ∈ M , as long as µ = −∞ or µ = ∞. Hence, to avoid unnecessary operations we start by checking if j ∈ M when µ > −∞ or µ < ∞; this is in contrast to the algorithms in [Kiw08a, Section 3] and [DWW12].…”

“…Note that we make use of information from earlier iterations when updating δ. When updating δ in [DWW12] information from earlier iteration is negligible hence some operations are repeated.…”