Optimization Methods Available in pSeven



General purpose methods (used in most algorithms below)

  • Determination of steepest/Quasi-Newton/Quadratically-Constrained direction and magnitude of optimal descent
  • Pool of one-dimensional line search methods
  • Specialized linear equations solver aimed to consider saddle-point systems
  • Universal problem transformations respecting complete independence of all representations and hence the parallelism of execution
Noisy data handling techniques
  • Needed to handle noise in problem observables, typically present in engineering applications.
  • Implemented at the lowest level to release upper-level algorithms (notably gradient-based) from problem noisiness.
Mathematical programming: Mixed-Integer Linear Programming (MILP) External solvers are used (current default is CBC/CLP)
Mathematical programming: Quadratic Programming (QP)
  • In-house developed implementation
  • Works for both definite/indefinite problems
  • Suitable for engineering applications and is a prime solver for internally generated subproblems
Mathematical programming: Unconstrained Non-Linear Programming (UNLP) Nonlinear Simplex, Nonlinear Conjugate Gradients, all with original implementation and appropriate adaptations for engineering settings.
  • Derivative-free Powell method, also in original implementation with proper consideration of engineering specifics
  • Quasi-Newton family of methods with various updating/search strategies
Mathematical programming: Constrained Non-Linear Programming (NLP)
  • Sequential Quadratic Programming with Filtering (no merit functions)
  • Sequential Quadratically Constrained Quadratic Programming with Filtering

All methods are adapted to handle engineering issues (noise, implicit constraints, etc.)

Constraint Satisfaction Problems (CSP) Works via reformulation of CSP in terms of auxiliary NLP, hence uses the above pool of NLP methods.
Multi-Objective Optimization: finding single solution 

In many cases, the whole Pareto frontier is not needed, and the nearest to current guess Pareto optimal solution is enough. Algorithms of this family contain in-house developed methods of multi-objective steepest/QN descent direction finding.

Multi-Objective Optimization Original gradient-based implementation avoids evaluations far from Pareto variety and in this sense is 'local' search algorithm.
Surrogate-Based Optimization (SBO): constrained single-objective problems

SBO is based upon a widely known probability improvement approach using Gaussian Processes(GP) to construct surrogate models. Algorithm is significantly different from other available implementations and is free of all the drawbacks of probability improvement methods. It utilizes custom GP modeling adapted for high problems dimensionality and other real-world applications issues. Additionally, it utilized an in-house developed DoE strategy, which respects as much feasibility domain of the problem as possible.

Surrogate-Based Optimization (SBO): constrained multi-objective problems Built on top of single-objective SBO, utilizes Chebyshev convolution to sequentially discover Pareto frontier.
Global Search Methods (excluding SBO)
  • Multi-start with SBO preconditioning: avoids too many initial points to be locally optimized via preliminary stage of SBO optimization.
  • Highly efficient in local searches with multiple initial guesses
  • Random Linkage: method belongs to multi-start family of methods. However, it uses various criteria to monitor problem multi-modality and to avoid unnecessary searches in simple problems. Due to relative expensiveness, it is applied to internally generated subproblems only
Robust Optimization
  • Original implementation of robust optimization supporting both expectation-like and probability-type observables.
  • Highly efficient due to the adaptive sample selection strategy.
  • Supports both direct and SBO-inspired approaches, so that it is suitable for expensive to evaluate problems.

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