Goal Programming, ReLU Networks and Endogenous Priorities: An α-Fair Unification of Scalarization

Abstract Weighted goal programming and the feedforward neural network both rank alternatives by a weighted aggregate of criteria plus an offset. We show that the resemblance is an identity: a linear achievement function is an affine neuron whose bias is minus the weighted aspiration level, weighted goal programming with one-sided deviations is a one-hidden-layer network of rectified linear units with nonnegative readout weights, and Chebyshev goal programming is the same layer followed by max-pooling. We then let the priorities depend on the alternative being evaluated through a power rule with a single exponent, which in network terms is softmax attention. Every fixed point of the induced weighted-sum iteration is a supported Pareto optimum and, on convex problems, a stationary point of the wellknown alpha-fair aggregate, so the exponent spans extreme-point seeking, the equal-weight sum, Nash bargaining and max–min compromise; below one, the fixed point is unique on convex problems and is computed by a Frank–Wolfe scheme. Maximizing the self-weighted score directly instead maximizes a Lehmer mean, which by a classical property respects Pareto dominance exactly when the exponent lies between zero and one, and selects a different compromise. Fixed points never reach unsupported Pareto points. Experiments on test problems and on the 2010 project portfolio of a U.S. Department of Defense conservation programme show that endogenous priorities recover the goal-programming portfolio without any weights being chosen, while the max–min portfolio is unsupported and, as the theory predicts, out of their reach.