Algorithms for Nonlinear Nonconvex Optimization under Uncertainty

研究人员将开发计算方法，寻求在并非所有信息都确切已知的情况下找到最佳决策。当未来事件无法准确预测或只能从有限的数据中估计数量时，就会出现这种情况。当使用计算机程序来模拟过程时，也会出现类似的设置。在这种情况下，由于底层数值算法的精度有限或由于模拟中固有的随机性，计算机输出可能是有噪声的。一个说明性的例子是太阳能或风能的部署：基于过去观测的电力需求估计是不确定的，天气预报模型是从初始条件的随机扰动开始的。现有的计算优化算法可以解决数据的不确定性，但目前的方法无法处理数学优化模型中复杂的非线性关系。这项研究将在两个方面克服这一限制。第一个研究项目将导致优化算法的问题，其中约束只需要满足给定的概率。这些方法将允许比目前最先进的技术更广泛的这些约束范围。第二个项目将通过明确处理输出噪声的性质来产生优化计算机模拟的方法。与现有方法相比，这些算法不会在由噪声引起的伪解处停滞不前。所有的新方法都将在软件中实现，并在实际问题中进行评估，并且将发展新的数学理论来证明这些方法的收敛性。在本项目中，将开发用于连续机会约束优化的新算法。在目前的方法中，目标函数和约束函数都要求是线性或凸的，并且由机会约束引起的非凸性要么通过保守的凸逼近来处理，要么通过组合分支定界枚举来处理离散公式的全局解。新方法将允许涉及非线性和非凸目标函数的问题陈述，并包括具有非凸概率约束的联合机会约束。这可以通过只寻求局部最优来实现，局部最优比全局最优更容易找到，但在实践中仍然很有价值。因此，从非凸非线性优化建立的技术可以建立和扩展。新的序贯二次机会约束规划框架需要引入新的机会约束信任域子问题解和机会约束惩罚函数的收敛理论，这些将在本项目中发展。PI还将为计算机模拟中由数值误差引起的确定性噪声的目标函数开发一种无导数优化方法。该方法基于高斯核卷积得到的光滑目标函数。新目标中的积分用蒙特卡罗样本平均近似逼近。自适应多重重要性采样允许重用在所有以前的迭代中计算的昂贵的函数计算。

The investigator will develop computational methods that seek to find optimal decisions when not all information is known exactly. This situation arises when future events cannot be predicted with high accuracy or when quantities can only be estimated from limited data. A similar setting occurs when computer programs are employed to simulate processes. In that case, the computer output can be noisy, due to limited precision of the underlying numerical algorithms or due to randomness inherent in the simulation. An illustrative example is the deployment of solar or wind energy: Electricity demand estimates based on past observations are uncertain, and weather forecast models are started from random perturbations of the initial conditions. Computational optimization algorithms exist that address data uncertainty, but current methods are not able to handle difficult nonlinear relationships in the mathematical optimization model. This research will overcome this limitation on two fronts. The first research project will result in optimization algorithms for problems in which constraints need to be satisfied only with a given probability. These methods will permit a much wider range of these constraints than the present state-of-the-art. The second project will produce methods that optimize computer simulations by explicitly addressing the nature of the output noise. In contrast to existing approaches, these algorithms will not stagnate at spurious solutions induced by the noise. All new methods will be implemented in software and evaluated on real-life problems, and new mathematical theory will be developed that proves the convergence of these methods.In this project, new algorithms for continuous chance-constrained optimization will be developed. In current approaches, the objective and constraint functions are required to be linear or convex, and the nonconvexity induced by the chance-constraints is handled either by conservative convex approximations or by the global solution of discrete formulations via combinatorial branch-and-bound enumeration. The new methods will permit problem statements that involve nonlinear and nonconvex objective functions and include joint chance constraints with nonconvex probabilistic constraints. This is made possible by seeking only local optima, which can be found more easily than global minima but are still highly valuable in practice. As a result, established techniques from nonconvex nonlinear optimization can be built upon and extended. The new sequential quadratic chance-constrained programming framework requires the introduction of new chance-constrained trust-region subproblem solvers and convergence theory for chance-constrained penalty functions which will be developed in this project. The PI will also develop a derivative-free optimization method for objective functions with deterministic noise caused by numerical error in computer simulations. The approach is based on a smoothed objective function obtained via convolution with a Gaussian kernel. The integral in the new objective is approximated by Monte-Carlo sample average approximation. Adaptive multiple importance sampling permits the reuse of the expensive function evaluations computed in all previous iterations.