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The price‐setting newsvendor problem is well studied in the literature. However, it is commonly assumed that retailers have complete demand information modeled as a function of price and random noise. In reality, a retailer may have very limited information on a demand model because a retailer who has exercised only a few prices does not have sufficient information to accurately estimate a demand model. This creates a gap between academic research and practical applications. In this study, we consider the price‐setting newsvendor problem in which the retailer knows the expected demand on a few exercised price points and the distribution of the random noise. Both additive and multiplicative demand models are studied. The retailer makes price and inventory decisions to minimize the maximum regret, which is defined as the difference between the expected profit based on limited demand information and that based on complete demand information. We show that this robust optimization problem can be reduced to a one‐dimensional optimization problem, and we derive the optimal price and inventory decisions. We also provide a demand learning policy that can reduce the minmax regret to any δ within O(1δ) steps. We also study the problem in which the retailer maximizes the worst‐case profit, and we obtain similar results. Extensive numerical studies show that our method has a great performance that dominates that of the regression method. [ABSTRACT FROM AUTHOR]

In this paper we consider a class of non-Lipschitz and non-convex minimization problems which generalize the $$L_2$$ - $$L_p$$ minimization problem. We propose an iterative algorithm that decides the next iteration based on the local convexity/concavity/sparsity of its current position. We show that our algorithm finds an $$\epsilon $$ -KKT point within $$O(\log {\epsilon ^{-1}})$$ iterations from certain initial points. The same result is also applied to the problem with general linear constraints under mild conditions. [ABSTRACT FROM AUTHOR]