阶跃优化是指目标函数或约束函数中含有一个阶跃函数的优化问题，这里阶跃函数是自变量大于0时取值为1、否则取值为0的函数。阶跃优化在机器学习、压缩感知、信号与图像处理、统计回归等领域有广泛应用，然而它是一个NP-难问题，通常做法是近似转化求解。本项目旨在开展阶跃优化本原模型理论与算法研究，具体内容包括：（1）通过分析阶跃函数的变分性质等，建立其原模型的最优性理论、对偶理论、罚函数理论等；（2）设计求解该模型的PG/ADMM算法和子空间牛顿算法等，使之具有全局收敛性、稳定性、快速性；（3）开发性能优异的求解器并应用到深度学习、1-bit压缩感知、支持向量机分类等实际问题。本项目的实施不仅能够解决重大需求中的一些实际问题以及技术瓶颈背后的核心科学问题，而且能够为解决非凸非连续优化等困难问题提供一条新途径，具有重要的科学意义和实用价值。

Sgn optimization is an optimization problem that the objective function or the constraint function involves an sgn function, where the sgn function returns 1 if the independent variable is a positive number and 0 otherwise. It has been extensively applied into a variety of fields ranging from machine learning, compressed sensing, signal and image processing to statistical regression. When it comes to solving the sgn optimization, a popular methodology is to seek for a relaxation due to its NP-hardness. This project aims to study the theory and algorithms for the original sgn optimization model in three aspects: (1) Establish the optimality theory, duality theory and penalty function theory of the original model through analyzing the variational properties of sgn function; (2) Design several efficient algorithms, such as PG/ADMM algorithm or subspace Newton algorithm, that are able to converge globally, stably and quickly; (3) Develop some powerful solvers based on those designed algorithms with abilities to solve practical problems arising from deep learning, 1-bit compressed sensing, support vector machine classification and so forth. The success of this project will make significant contributions theoretically and practically because it not only could break through the real-world problems and the technical bottleneck stemmed from the core issue of the original sgn optimization, but also could give a new insight into dealing with non-convex and discontinuous optimization problems.