On solving nonsmooth retail portfolio maximization problems using active signature methods

The retail industry is governed by crucial decisions on inventory management, discount offers like promotions and stock clearing as so-called markdowns, presenting two sets of optimization problems. The former is an estimation problem, where the underlying objective is to predict the coefficients of demand (sales) elasticity with respect to product prices. The latter is the dynamic revenue maximization problem, which takes in the coefficients of demand as inputs. While both tasks present nonsmooth optimization problems, the latter is a challenging nonlinear problem in massive dimensions. This is further subject to constraints on inventory, inter-product relationships, and price bounds. Traditional approaches to solve such problems relied on using reformulations and approximations, thereby leading to potentially suboptimal solutions. In this work, we retain the nonsmooth structure generated by the $$\max$$ type (or equivalently absolute value type) function and solve the resulting problem in its abs-quadratic form, i.e., in a quadratic matrix-vector-product based representation including linear arguments in the abs-evaluation. Subsequently, we present an adaptation of the Constrained Active Signature Method (CASM) that explicitly exploits this abs-quadratic structure of the problem yielding the Quadratic Constrained Active Signature Method (QCASM). In the process, we also guarantee convexity of the objectives under some mild realistic assumptions on the market demand and structure. Two real world retail examples (UK and US market data from 2017-2019) and one simulated use-case are studied from an empirical standpoint. Numerical results demonstrate good performance of QCASM and further show that such solvers can be used significantly by the retail science community in the future.