A parameter-free multiplier method for constrained minimization problems

AbstractThis paper is concerned with the development of a parameter-free method, closely related to penalty function and multiplier methods, for solving constrained minimization problems. The method is developed via the quadratic programming model with equality constraints. The study starts with an investigation into the convergence properties of a so-called “primal-dual differential trajectory”, defined by directions given by the direction of steepest descent with respect to the variables x of the problem, and the direction of steepest ascent with respect to the Lagrangian multipliers λ, associated with the Lagrangian function. It is shown that the trajectory converges to a stationary point (x*, λ*) corresponding to the solution of the equality constrained problem. Subsequently numerical procedures are proposed by means of which practical trajectories may be computed and the convergence of these trajectories are analyzed. A computational algorithm is presented and its application is illustrated by means of simple but representative examples. The extension of the method to inequality constrained problems is discussed and a non-rigorous argument, based on the Kuhn—Tucker necessary conditions for a constrained minimum, is put forward on which a practical procedure for determining the solution is based. The application of the method to inequality constrained problems is illustrated by its application to a couple of simple problems

The spherical quadratic steepest descent (SQSD) method for unconstrained minimization with no explicit line searches

AbstractA very simple gradient only algorithm for unconstrained minimization is proposed that, in terms of storage requirement and computational efficiency, may be considered as an alternative to the conjugate gradient line search methods for large problems. The method effectively applies the steepest descent method to successive simple (spherical) quadratic approximations of the objective function in such a way that no explicit line searches are performed in solving the minimization problem. It is shown that the method is convergent when applied to general positive-definite quadratic functions. The method is tested by its application to some standard and other test problems. On the evidence presented, the new method, called the SQSD algorithm, appears to be reliable and stable, and very competitive compared to the well-established Fletcher-Reeves and Polak-Ribiere conjugate gradient methods. In particular, it does very well when applied to extremely ill-conditioned problems

A randomised controlled trial comparing oxytocin and oxytocin + ergometrine for prevention of postpartum haemorrhage at caesarean section

BACKGROUND. Globally 166 000 women die annually as a result of obstetric haemorrhage. More than 50% of these deaths occur in sub- Saharan Africa. Uterine atony is the commonest cause of severe postpartum haemorrhage (PPH). Bleeding at or after caesarean section (CS) is responsible for >30% of maternal deaths due to obstetric haemorrhage in South Africa (SA). OBJECTIVE. To compare oxytocin alone with oxytocin + ergometrine in terms of primary prophylaxis for PPH at the time of CS. METHODS. This was a double-blind randomised controlled interventional study comparing oxytocin with oxytocin + ergometrine administered during CS. Patients were randomised to receive oxytocin alone intravenously as a bolus or oxytocin + ergometrine intramuscularly, with the placebo being an injection of sterile water. The study population consisted of women undergoing CS at Kalafong Provincial Tertiary Hospital in Atteridgeville, Gauteng, SA. RESULTS. Five hundred and forty women were randomised and data for 416 women, of whom 214 received oxytocin and 202 oxytocin + ergometrine, were available for analysis. In the oxytocin group 19 women (8.9%) required blood transfusion, compared with seven (3.5%) in the oxytocin + ergometrine group (p=0.01; relative risk = 2.78; 95% confidence interval 1.21 - 6.4). There were no statistically significant differences in the mean estimated visual and mean calculated blood loss. CONCLUSIONS. The overall need for blood transfusion was significantly reduced by about two-thirds in women receiving the oxytocin + ergometrine combination. Consideration should be given to using oxytocin + ergometrine for prophylaxis of PPH at CS.http://www.samj.org.zaam2016Obstetrics and Gynaecolog

A unified feasible direction interior approach to the minimization of linear and general objective functions subject to linear constraints

A modification of Snyman's interior feasible direction method for linear programming is proposed and the method is also extended to problems where the objective function is non-linear. The method attempts to identify the optimal bounding set of active constraints. In the modified algorithm the successive interior steps in the identifying cycle are no longer constrained to be in the place of constant objective function value, but are computed to ensure improvement in the objective function for any non-zero step taken within the cycle. The method is also extended to non-linear objective functions by allowing for line searches within the interior and along bounding hypersurfaces. A formal unified algorithm is presented and the method is illustrated by its successful application to a number of simple problems from different categories