IE680_KuoHao - Stochastic Trust Region GradientFree Method...

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Stochastic Trust Region Gradient- Free Method (STRONG) -A Response-Surface-Based Algorithm in Stochastic Optimization via Simulation Kuo-Hao Chang Advisor: Hong Wan School of Industrial Engineering, Purdue University Acknowledgement: The project was partially supported by grant from Naval Postgraduate School. Purdue University
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2 Outline Background Problem Statement Literatures Review STRONG Preliminary Numerical Evaluations Future Research
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3 Background Stochastic Optimization The minimization (or maximization) of a function in the presence of randomness Optimization via Simulation: No explicit form of the objective function (only observations from simulation), function evaluations are stochastic and usually computationally expensive. Applications Investment portfolio optimization, production planning, traffic control etc.
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4 Problem Statement (I) Consider the unconstrained continuous minimization problem The response can only be observed by : randomness defined in the probability space : the noisy term showing dependence on x x ) x ( L ) , x ( Q ε ϖ + = P) F, , ( x )) , x ( Q ( E min arg x
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5 Problem Statement (II) Given : a simulation oracle of capable generating s.t. Strong Law of Large Numbers hold for every Find : a local minimizer , i.e., find having a neighborhood such that every satisfies * x ) , x ( Q ϖ x * x ) x ( V * ) x ( V x * ) x ( L ) x ( L *
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6 Problem Assumptions For 1. 2. For the underlying function 1. is bounded below and twice differentiable for every 2. ) , 0 ( N ~ 2 x x σ ε < 2 x 2 x x sup and unknown is x ) x ( L 1 1 1 1 H(x) , ) ( such that , , , β α 5 2200 x L x ) x ( L x
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7 Literatures Review (Fu, 1994; Fu 2002) Methodology Efficient Convergent Automated Stochastic Approximation Usually No Yes Human tuning Sample-Path Optimization Usually Yes Yes Yes Response Surface Methodology (RSM) Yes No No Other Heuristic Methods (e.g. Genetic Algorithm, Tabu Search etc.) Yes Usually No Yes
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8 Proposed Work A RSM-based method with convergence property (combining the trust region method for deterministic optimization with the RSM) Does not require human involvement Appropriate DOE to handle high-dimensional problems (on-going work)
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9 Response Surface Methodology Stage I Employ a proper experimental design Fit a first-order model Perform a line search Move to a better solution Stage II (when close to the optimal solution) Employ a proper experimental design Fit a second-order model Find the optimal solution
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10 RSM (Mongomery, 2001)
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11 Deterministic Trust Region Framework (Conn et al. 2000) Suppose we want to minimize a deterministic objective function f(x) Step 0 : Given an initial point ,an initial trust-region radius , and some constants satisfy and set Step 1 : Compute a step within the trust region that “sufficiently reduces” the local model constructed by Taylor expansion (to second-order) around
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This note was uploaded on 11/13/2010 for the course ISE 680 taught by Professor Santanu during the Spring '10 term at Purdue University Calumet.

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IE680_KuoHao - Stochastic Trust Region GradientFree Method...

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