From Insight to Allocation: Applying Linear and Nonlinear Optimizers for Daily Budget Moves

Daily budget reallocation in digital and performance marketing is often executed as a sequence of heuristics, despite the fact that the underlying decision is structurally an economic allocation problem under constraints.This abstract formalizes the "insight to allocation" pipeline that links empirical inference about incremental returns to an auditable, solver-based decision rule for day-to-day budget moves.The proposed framing treats each channel or activity as an investment option with a response function that maps spend to expected incremental outcome, while governance requirements are expressed as feasibility constraints representing total budget, minimum commitments, capacity ceilings, risk limits, and operational change frictions.Within this architecture, linear programming is positioned as a transparent baseline: when marginal effects are approximately constant over the adjustment range, the allocation can be written as a linear objective with linear constraints, and dual variables provide economically meaningful shadow prices that support managerial interpretation and constraint valuation.The abstract then motivates nonlinear optimization as soon as diminishing returns, carryover, and lagged effects become material.Concave saturation curves and state dynamics such as adstock produce objectives that can be convex, nonconvex, or simulation-defined, shifting the solver choice toward nonlinear programming, mixedinteger nonlinear programming, or derivative-free and evolutionary methods.A key contribution is the explicit mapping from measurement artifacts, for example marginal effects and uncertainty intervals, into objective coefficients and constraint bounds, enabling robust or stochastic formulations when parameters drift.The resulting perspective reframes daily budget control as constrained capital allocation rather than tactical "tweaks," emphasizing that optimality is conditional on declared objectives, constraint libraries, and friction prices.By separating estimation, translation, optimization, and feedback into modular steps, the approach supports reproducibility, auditability, and clearer economic narratives about why budgets move.The abstract concludes that the practical choice between linear and nonlinear optimizers is not a question of sophistication, but of structural fit: linear models maximize interpretability and speed under near-linear response, while nonlinear models preserve economic realism under saturation and carryover, provided governance constraints are encoded to prevent operationally unsafe "optimal" solutions.