In this paper, we present a penalty function with objective parameters for inequality constrained optimization problems. We prove that this type of penalty functions has good properties for helping to solve inequality constrained optimization problems. Moreover, based on the penalty function, we develop an algorithm to solve the inequality constrained optimization problems and prove its convergence under some conditions. Numerical experiments show that we can obtain a satisfactorily approximate solution for some constrained optimization problems as the same as the exact penalty function.

To manage the risk arising from uncertainty in market demand, this paper introduces the Conditional Value-at-Risk (CVaR) measure into the decision framework of the newsvendor who aims to minimise his opportunity loss. It is found under the CVaR measure that the newsvendor’s optimal order quantity is increasing in the confidence level when the understock loss is bigger than the overstock loss. This implies that an over-ordering may be even more caused by the newsvendor’s risk aversion about opportunity loss than risk seeking behaviour. Under this optimal order quantity, it is proved that the newsvendor’s expected profit and expected opportunity loss are decreasing and increasing in the confidence level, respectively. Furthermore, some management insights are presented to facilitate the risk management of the newsvendor model.

Biconvex programming (or inequality constrained biconvex optimization) is an important model in solving many engineering optimization problems in areas like machine learning and signal and information processing. In this paper, the partial exactness of the partial optimum for the penalty function of biconvex programming is studied. The penalty function is partially exact if the partial Karush–Kuhn–Tucker (KKT) condition is true. The sufficient and necessary partially local stability condition used to determine whether the penalty function is partially exact for a partial optimum solution is also proven. Based on the penalty function, an algorithm is presented for finding a partial optimum solution to an inequality constrained biconvex optimization, and its convergence is proven under some conditions.