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Optimization:
- parameter space, x
- find maximun (or minimum) of objective function, f(x)
- satisfying certian constraints (optional)
- gi(x) = 0
- hj(x) < 0 (or > 0)
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Categorize by
- objective function; linear v. non-linear; convex, concave
- constraint equations: linear, non-linear; equality, inequality; convex, concave
- search space: continuous, discrete
- search method: e.g., gradient descent, random
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Other links
first projects
- Kevin - matlab
- Alex - some optimization problem from Dr. Shen
- Zhaohui - HQP software
- Ben - space-time particle
- Jae - simple optimization problem
- KC - NLOPT?
- Zhili - opt packages on spacetime particle
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Optimization categories
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Constrained Optimization (pdf of slides)
- ACM Portal breakdown
- Constrained optimization
- Convex programming
- Global optimization
- Gradient methods
- Inter programming
- Least squares methods
- Linear programming
- Nonlinear programming
- Quadratic programming methods
- Simulated annealing
- Stochastic programming
- Unconstrained optimization
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Wikipedia's optimization breakdown
- Convex programming
- Integer programming
- Quadratic programming
- Nonlinear programming
- Stochastic programming
- Robust programming
- Combinatorial optimization
- Infinite-dimensional optimization
- Heuristic algorithms
- Constraint satisfaction
- Disjunctive programming
- Trajectory optimization
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Methods commonly encountered
- Linear function, linear constraints
- Linear Programming
- Simplex Method
- Non-linear optimization
- Convex problems
- unconstrained
- Analytic first derivative = 0
- Iterative methods: descent methods, e.g., Gradient descent, Congugate gradient method, Powell's method, line search
- Quadratic Programming
- Sequential quadratic programming
- constrained
- Probabalistic algorithms
- Genetic algorithms
- Simulated Annealing
- Search methods
- Dynamic programming
- Viterbi algorithm
- Stochastic algorithms
- Monte Carlo methods
- Particle Swarm Optimization
- Discrete Methods
- integer programming
- scheduling problems
- knapsack problems
- network flow problems
- State estimation
- Kalman filter
- Condensation
Associated techniques often encountered
- dimensionality reduction
- PCA
- LU decomposition
- Isomap
- factor analysis
- linear discriminant analysis
- ...
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Optimization Approaches - web pages
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Optimization Packages and Software - web pages
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Sample problems
Some of these are toy problems some are complex.
Some of these are fabricated problems, some are useful.
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Given a cost function of object parameters, minimize the cost incurred over a time interval using constraints that the object is at location A at time t=0 and at location B at t=1. For example, the cost function might be a linear function of force applied to the object. Assum gravity and sticky collisions with the groundplane.
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Given a few measurements of a human figure (e.g., height, weight, chest circumfrence) process a database of human figures to select a figure that minimizes distance from the measurements.
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Given a database of motions, stitch together the best sequence of motions, with lowest transition cost and minimizes errors from path constraints.
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Given a family of articulated linkages, find the one the travels the greatest distance when the joint is repeatedly articulated.
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Given an articulated human figure, manimize the sum of joint torques that produce the most effective walking behavior.
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Examples from the literature
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