Solving large-scale nonlinear programming problems by constraint partitioning

Abstract. In this paper, we present a constraint-partitioning approach for finding local optimal solutions of large-scale mixed-integer nonlinear programming problems (MINLPs). Based on our observation that MINLPs in many engineering applications have highly structured constraints, we propose to partition these MINLPs by their constraints into subproblems, solve each subproblem by an existing solver, and resolve those violated global constraints across the subproblems using our theory of extended saddle points. Constraint partitioning allows many MINLPs that cannot be solved by existing solvers to be solvable because it leads to easier subproblems that are significant relaxations of the original problem. The success of our approach relies on our ability to resolve violated global constraints efficiently, without requiring exhaustive enumerations of variable values in these constraints. We have developed an algorithm for automatically partitioning a large MINLP in order to minimize the number of global constraints, an iterative method for determining the optimal number of partitions in order to minimize the search time, and an efficient strategy for resolving violated global constraints. Our experimental results demonstrate significant improvements over the best existing solvers in terms of solution time and quality in solving a collection of mixed-integer and continuous nonlinear constrained optimization benchmarks.

Alcohol and breast cancer in women: A pooled analysis of cohort studies

Objective. - To assess the risk of invasive breast cancer associated with total and beverage-specific alcohol consumption and to evaluate whether dietary and nondietary factors modify the association. Data Sources. - We included in these analyses 6 prospective studies that had at least 200 incident breast cancer cases, assessed long-term intake of food and nutrients, and used a validated diet assessment instrument. The studies were conducted in Canada, the Netherlands, Sweden, and the United States. Alcohol intake was estimated by food frequency questionnaires in each study. The studies included a total of 322 647 women evaluated for up to 11 years, including 4335 participants with a diagnosis of incident invasive breast cancer. Data Extraction. - Pooled analysis of primary data using analyses consistent with each study's original design and the random-effects model for the overall pooled analyses. Data Synthesis. - For alcohol intakes less than 60 g/d (reported by >99% of participants), risk increased linearly with increasing intake; the pooled multivariate relative risk for an increment of 10 g/d of alcohol (about 0.75-1 drink) was 1.09 (95% confidence interval [CI], 1.04-1.13; P for heterogeneity among studies, .71). The multivariate- adjusted relative risk for total alcohol intakes of 30 to less than 60 g/d (about 2-5 drinks) vs nondrinkers was 1.41 (95% CI, 1.18-1.69). Limited data suggested that alcohol intakes of at least 60 g/d were not associated with further increased risk. The specific type of alcoholic beverage did not strongly influence risk estimates. The association between alcohol intake and breast cancer was not modified by other factors. Conclusions. - Alcohol consumption is associated with a linear increase in breast cancer incidence in women over the range of consumption reported by most women. Among women who consume alcohol regularly, reducing alcohol consumption is a potential means to reduce breast cancer risk