Feasible Point Optimization Methods for Design and Other Engineering Applications

优化问题遍及人类努力的所有领域。这类问题通常包括在满足某些约束条件下最小化或最大化的目标函数（成本、利润、精度、重量、时间、燃料、耐久性等）。除了最简单的情况，有效解决这些问题都需要使用“数值”优化工具。本基金为开发、分析、实施和测试具有大量不等式约束的约束优化问题的数值算法提供资金。这类问题在工程应用中很常见。寻求具有较强的理论收敛性和较高的实际效率的算法。重点关注具有“可行性”和“单调下降”的算法；即，构造一个迭代序列（对解的越来越好的估计），其性质是(i)所有迭代都满足不等式约束，（ii）目标函数值在迭代中单调提高。可行性和单调下降法具有广泛的应用价值。特别是，在PI之前开发和实现算法的工作背景下，这些特性得到了从癌症研究到化学工程，到集成电路设计，到放射学等领域的用户的好评。此外，在工程和科学的许多应用中，具有大量不等式约束的问题是常见的。因此，可以预见的是，就像PI小组以前开发的算法和软件一样，如果成功的话，将从拟议的研究中得出的算法和软件将在广泛的应用领域产生重大影响。

Optimization problems pervade all areas of human endeavor. Such problems generally include an objective function to be minimized or maximized (cost, profit, accuracy, weight, time, fuel, durability, etc.) subject to satisfaction of certain constraints. In all but the simplest cases, effectively tackling these problems requires the use of "numerical" optimization tools. The present grant provides funding for developing, analyzing, implementing, and testing numerical algorithms for the solution of constrained optimization problems with a large number of inequality constraints. Such problems are common in engineering applications. Algorithms are sought that exhibit strong theoretical convergence properties as well as high practical efficiency. Attention is focused on algorithms that enjoy feasibility" and "monotone descent"; i.e., that construct a sequence of iterates (increasingly better estimates of the solution) with the property that (i) all iterates satisfy the inequality constraints, and (ii) the objective function value improves monotonically from iteration to iteration.Feasibility and monotone descent are of value in a broad range of application areas. In particular, in the context of the PI's previous work on developing and implementing algorithms, such properties have been acclaimed by users in areas ranging from cancer research to chemical engineering, to design of integrated circuits, to radiology, to mention but a few. Further, problems with a very large number of inequality constraints are common in many applications ranging over engineering and the sciences. Accordingly, it is anticipated that, as has been the case with previous algorithms and software developed by the PI's group, the algorithms and software that will come out of the proposed research if it is successful, will have a significant impact in a wide range of application areas.