A Non-Smooth Analysis Approach for Optimization of Complex Industrial Processes
Keywords:
Non-smooth optimization, clarke subdifferential, generalized gradient, proximal stabilization, constrained optimization, industrial process optimization.Abstract
Non-smooth objective functions may occur in industrial optimization problems due to the existence of thresholds, switches, piecewise objective functions, absolute value terms, and bounds on the feasible set. Gradient-based approaches may lose their efficiency due to the lack of differentiability assumption. In this paper, a mathematical framework for constrained non-smooth optimization is developed. The theory includes a composite representation of the objective function in terms of continuously differentiable and locally Lipschitz non-smooth components, Clarke generalized derivatives, minimum-norm subgradients rule, geometry of tangent and normal cones, proximal stabilization, and adaptive step-size choice. A deterministic minimum norm selection criterion is used to deal with a multi-valued generalized subdifferential. Feasibility is achieved via Euclidean projection. The proximal term makes the algorithm stable in the vicinity of non-differentiable points. The theoretical properties include the existence of an optimum solution, sufficient descent, objective monotonicity, boundedness of the sequence, and generalized stationarity of its accumulation points under certain regularity assumptions. Numerical performance is analyzed on three test cases: absolute-value, maximum-type, and L1-regularized composite benchmark functions and an engineering-oriented problem of the industrial process optimization in terms of production cost, energy consumption, and threshold penalties. It is shown that the proposed framework can retain non-smooth features without complete differentiable approximation.
Downloads
Downloads
Published
Issue
Section
License
You are free to:
- Share — copy and redistribute the material in any medium or format for any purpose, even commercially.
- Adapt — remix, transform, and build upon the material for any purpose, even commercially.
- The licensor cannot revoke these freedoms as long as you follow the license terms.
Under the following terms:
- Attribution — You must give appropriate credit , provide a link to the license, and indicate if changes were made . You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- No additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.
Notices:
You do not have to comply with the license for elements of the material in the public domain or where your use is permitted by an applicable exception or limitation .
No warranties are given. The license may not give you all of the permissions necessary for your intended use. For example, other rights such as publicity, privacy, or moral rights may limit how you use the material.
