- Chance constrained programming
- Chance-constrained portfolio selection
- Stochastic programming
- András Prékopa
- Abraham Charnes
- Robust fuzzy programming
- William W. Cooper
- Scenario optimization
- Register allocation
- Evolutionary algorithm
- Chance constrained programming - Wikipedia
- Chance-constraint method - Cornell University Computational ...
- Chance constrained optimization - Stanford University
- CHANCE-CONSTRAINED PROGRAMMING - Carnegie …
- Lecture 24: Robust Optimization: Chance Constraints
- Chance Constrained Programming and Its Applications to
- Chance Constrained Programming - GitHub Pages
- Chance constraints and distributionally robust optimization
- Chance-Constrained Programming: Joint and Individual …
- Chance-Constrained Programming - an overview - ScienceDirect
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Chance Constrained Programming (CCP) is a mathematical optimization approach used to handle problems under uncertainty. It was first introduced by Charnes and Cooper in 1959 and further developed by Miller and Wagner in 1965. CCP is widely used in various fields, including finance, engineering, and operations research, to optimize decision-making processes where certain constraints need to be satisfied with a specified probability.
Theoretical Background
Chance Constrained Programming involves the use of probability and confidence levels to handle uncertainty in optimization problems. It distinguishes between single and joint chance constraints:
Single Chance Constraints: These constraints ensure that each individual constraint is satisfied with a certain probability.
Joint Chance Constraints: These constraints ensure that all constraints are satisfied simultaneously with a certain probability.
Mathematical Formulation
A general chance constrained optimization problem can be formulated as follows:
min
f
(
x
,
u
,
ξ
)
s.t.
g
(
x
,
u
,
ξ
)
=
0
,
Pr
{
h
(
x
,
u
,
ξ
)
≥
0
}
≥
α
{\displaystyle \min f(x,u,\xi ){\text{s.t. }}g(x,u,\xi )=0,\Pr\{h(x,u,\xi )\geq 0\}\geq \alpha }
Here,
f
{\displaystyle f}
is the objective function,
g
{\displaystyle g}
represents the equality constraints,
h
{\displaystyle h}
represents the inequality constraints,
x
{\displaystyle x}
represents the state variables,
u
{\displaystyle u}
represents the control variables,
ξ
{\displaystyle \xi }
represents the uncertain parameters, and
α
{\displaystyle \alpha }
is the confidence level.
Common objective functions in CCP involve minimizing the expected value of a cost function, possibly combined with minimizing the variance of the cost function.
Solution Approaches
To solve CCP problems, the stochastic optimization problem is often relaxed into an equivalent deterministic problem. There are different approaches depending on the nature of the problem:
Linear CCP: For linear systems, the feasible region is typically convex, and the problem can be solved using linear programming techniques.
Nonlinear CCP: For nonlinear systems, the main challenge lies in computing the probabilities and their gradients. These problems often require nonlinear programming solvers.
Dynamic Systems: Dynamic systems involve time-dependent uncertainties, and the solution approach must account for the propagation of uncertainty over time.
Practical Applications
Chance constrained programming is used in engineering for process optimisation under uncertainty and production planning and in finance for portfolio selection. It has been applied to renewable energy integration, generating flight trajectory for UAVs, and robotic space exploration.
= Process Optimization Under Uncertainty
=CCP is used in chemical and process engineering to optimize operations considering uncertainties in operating conditions and model parameters. For example, in optimizing the design and operation of chemical plants, CCP helps in achieving desired performance levels while accounting for uncertainties in feedstock quality, demand, and environmental conditions.
= Production Planning and Operations
=In production planning, CCP can optimize production schedules and resource allocation under demand uncertainty. A typical problem formulation involves maximizing profit while ensuring that production constraints are satisfied with a certain probability.
= Chance-Constrained Portfolio Selection
=Chance-constrained portfolio selection is an approach to portfolio selection under loss aversion which is based on CCP. The goal is to maximize expected returns while ensuring that the portfolio's risk (e.g., variance or downside risk) stays within acceptable levels with a certain probability. This approach allows investors to consider the uncertainty in asset returns and make more informed investment decisions.
References
Kata Kunci Pencarian:
Chance-constrained programming. | Download Scientific Diagram
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(PDF) A Two-Stage Optimal Scheduling Model of Microgrid Based on Chance ...
Chance-constrained programming. | Download Scientific Diagram
Solved 3. Compare and contrast the chance constrained and | Chegg.com
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(PDF) Chance Constrained Programming for Optimal Power Flow Under ...
(PDF) Solving Chance-Constrained Stochastic Programs via Sampling and ...
chance constrained programming
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Chance constrained programming - Wikipedia
Chance Constrained Programming (CCP) is a mathematical optimization approach used to handle problems under uncertainty. It was first introduced by Charnes and Cooper in 1959 and further developed by Miller and Wagner in 1965.
Chance-constraint method - Cornell University Computational ...
Dec 15, 2021 · The chance-constraint method of optimization programming is a process for working with random parameters within a problem while guaranteeing a certain performance. Uncertain variables in a project lead to questions regarding reliability and risk which make for difficulties in determining the most likely result.
Chance constrained optimization - Stanford University
Markov chance constraint bound • taking φ(u) = (u+1)+ gives Markov bound: for any αi > 0, Prob(fi(x,ω) > 0) ≤ E(fi(x,ω)/αi +1)+ • convex approximation constraint Eαi(fi(x,ω)/αi +1)+ ≤ αi(1−η) can be written as E(fi(x,ω)+αi)+ ≤ αi(1−η) • we can optimize over x and αi ≥ 0 EE364A — Chance Constrained Optimization 13
CHANCE-CONSTRAINED PROGRAMMING - Carnegie …
The problem of stochastic (or better, chance-constrained) programming is here defined as follows: Select certain random variables as functions of random variables with known distributions in such a manner as (a) to maximize a func tional of both classes of random variables subject to (b) constraints on these variables which must be maintained at...
Lecture 24: Robust Optimization: Chance Constraints
Chance constraints are a probabilistic way of handling probabilistic uncertainty. We would like to convert chance constraints into robustness constraints, which are easier to deal with.
Chance Constrained Programming and Its Applications to
Chance Constrained Programming belongs to the major approaches for dealing with random parameters in optimization problems.
Chance Constrained Programming - GitHub Pages
Single Constraint Easy Case † The situation in the single constraint case is somewhat more simple. † Suppose again that Ti(!) = Ti is constant. (The ith row of the technology matrix is constant, and we wish to enforce...) P(Tix ‚ hi(!)) = F (Tix) ‚ …
Chance constraints and distributionally robust optimization
Chance constraints non-convex problem minimize f 0(x) subject to Prob(fi(x,U) > 0) ≤ ǫ, i = 1,...,m safe approximation: find convex gi: Rn → R such that gi(x) ≤ 0 implies Prob(fi(x,U) > 0) ≤ ǫ • sufficient condition: find set U such that Prob(U ∈ U) ≥ 1−ǫ set gi(x) = sup u∈U fi(x,u) EE364b, Stanford University 1
Chance-Constrained Programming: Joint and Individual …
Feb 16, 2023 · A number of reviews and surveys on chance-constrained programming exist. For textbooks and works of an introductory nature, see and references cited within. Optimization models with chance constraints are hard to solve directly, …
Chance-Constrained Programming - an overview - ScienceDirect
Chance-Constrained Programming is a method that introduces uncertainty into design optimization problems by ensuring that certain constraints are satisfied with a specified probability or reliability level.