DTU CEE Summer School 2020 - Distributed Optimization · 2019. 6. 20. · 3 DTU Electrical...
Transcript of DTU CEE Summer School 2020 - Distributed Optimization · 2019. 6. 20. · 3 DTU Electrical...
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Distributed Optimization
Lecture 1: Optimization problems with decomposable structure
Jalal Kazempour
26 June 2015June 21, 2019
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26 June 2015DTU Electrical Engineering, Technical University of Denmark2 Jalal Kazempour
Agenda
Lecture Topic Time period
1 Optimization problems with decomposable structures(and Exercise 1 including discussion)
9:15 – 10:00
Break 10:00 – 10:10
2 Benders’ decomposition (and Exercise 2) 10:10 – 11:00
Break 11:00 – 10:10
Discussion on Exercise 2 11:10 – 11:30
3 Part 1: Lagrangian relaxation 11:30 – 12:00
Lunch 12:00 – 13:30
Side Talk (Introduction to MOSEK by M. Adamaszek) 13:30 – 14:00
3 Part 2: Augmented Lagrangian relaxation 14:00 – 14:30
Break 14:30 – 14:40
4 Exchange and consensus ADMM (and Exercise 3) 14:40 – 15:30
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Learning objectives
After Lecture 1, you are expected to be able to:
• Explain the need for decomposition
• Identify whether each optimization problem isdecomposable or not (if so, how?)
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Decomposition: main idea
Original (non‐decomposed) optimization problem with decomposable structure
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26 June 2015DTU Electrical Engineering, Technical University of Denmark5 Jalal Kazempour 2/26
Decomposition: main idea
Decomposition
Original (non‐decomposed) optimization problem with decomposable structure
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26 June 2015DTU Electrical Engineering, Technical University of Denmark6 Jalal Kazempour 2/26
Decomposition: main idea
Decomposedoptimization problem 1
Decomposition
Decomposedoptimization problem 2
Decomposedoptimization problem n
Each decomposed problem is easier to solve than the original(non‐decomposed) problem!
Original (non‐decomposed) optimization problem with decomposable structure
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Decomposition: motivation
• Why do we need decomposition in energy systems?
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Decomposition: motivation
• Why do we need decomposition in energy systems?
Operational problems (e.g., unit commitment)
Planning problems (e.g., investment)
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26 June 2015DTU Electrical Engineering, Technical University of Denmark9 Jalal Kazempour 3/26
Decomposition: motivation
• Why do we need decomposition in energy systems?
Operational problems (e.g., unit commitment)
Planning problems (e.g., investment)
Need to be computationally tractable
Need to be solved in a specific time period
Need to be computationally tractable
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26 June 2015DTU Electrical Engineering, Technical University of Denmark10 Jalal Kazempour 3/26
Decomposition: motivation
• Why do we need decomposition in energy systems?
Operational problems (e.g., unit commitment)
Planning problems (e.g., investment)
Need to be computationally tractable
Need to be solved in a specific time period
Need to be computationally tractable
Any other reason for decomposition?
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26 June 2015DTU Electrical Engineering, Technical University of Denmark11 Jalal Kazempour 4/26
Decomposable structure
Optimization problems with decomposable structure:
Problems with complicating variable(s):
Problems with complicating constraint(s):
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26 June 2015DTU Electrical Engineering, Technical University of Denmark12 Jalal Kazempour 4/26
Decomposable structure
Optimization problems with decomposable structure:
Problems with complicating variable(s):
The original problem is decomposed if the complicatingvariables are fixed to given values!
Problems with complicating constraint(s):
The original problem is decomposed if the complicatingconstraints are relaxed (removed)!
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26 June 2015DTU Electrical Engineering, Technical University of Denmark13 Jalal Kazempour 4/26
Decomposable structure
Optimization problems with decomposable structure:
Problems with complicating variable(s):
The original problem is decomposed if the complicatingvariables are fixed to given values!
Problems with complicating constraint(s):
The original problem is decomposed if the complicatingconstraints are relaxed (removed)!
In the literature, “complicating” variables/constraints arealso called as “coupling” variables/constraint!
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26 June 2015DTU Electrical Engineering, Technical University of Denmark14 Jalal Kazempour 5/26
Features of decomposition techniques
• Iterative solution techniques
• Original optimization problem decomposes to: A single master problem (not necessarily an optimization
problem)
A set of subproblems
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Decomposable structures
Optimization problems with complicating constraint(s)
Subject to
Example: a linear programming (LP) as the original problem
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26 June 2015DTU Electrical Engineering, Technical University of Denmark16 Jalal Kazempour 6/26
Decomposable structures
Optimization problems with complicating constraint(s)Example: a linear programming (LP) as the original problem
Subject to
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26 June 2015DTU Electrical Engineering, Technical University of Denmark17 Jalal Kazempour 6/26
Decomposable structures
Optimization problems with complicating constraint(s)Example: a linear programming (LP) as the original problem
Subject to Constraints including only variables
Constraint including only variables
Constraints including only variables
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26 June 2015DTU Electrical Engineering, Technical University of Denmark18 Jalal Kazempour 6/26
Decomposable structures
Optimization problems with complicating constraint(s)Example: a linear programming (LP) as the original problem
Subject to Constraints including only variables
Constraint including only variables
Constraints including only variables
This is a complicating constraint: If relaxed (removed), then the original problem decomposes to three smaller optimization problems (subproblems)!
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Decomposable structures
Optimization problems with complicating constraint(s)Example: a linear programming (LP) as the original problem
Subproblem 1:
Subject to
Constraints including only variables
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26 June 2015DTU Electrical Engineering, Technical University of Denmark20 Jalal Kazempour 8/26
Decomposable structures
Optimization problems with complicating constraint(s)Example: a linear programming (LP) as the original problem
Subproblem 2:
Subject to
Constraint including only variables
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26 June 2015DTU Electrical Engineering, Technical University of Denmark21 Jalal Kazempour 9/26
Decomposable structures
Optimization problems with complicating constraint(s)Example: a linear programming (LP) as the original problem
Subproblem 3:
Constraints including only variables
Subject to
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26 June 2015DTU Electrical Engineering, Technical University of Denmark22 Jalal Kazempour 10/26
Decomposable structures
Optimization problems with complicating constraint(s)
Subject to
Complicating constraint
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26 June 2015DTU Electrical Engineering, Technical University of Denmark23 Jalal Kazempour 11/26
Decomposable structures
Optimization problems with complicating variable(s)Example: a linear programming (LP) as the original problem
Subject to
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26 June 2015DTU Electrical Engineering, Technical University of Denmark24 Jalal Kazempour 11/26
Decomposable structures
Optimization problems with complicating variable(s)Example: a linear programming (LP) as the original problem
Subject to
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26 June 2015DTU Electrical Engineering, Technical University of Denmark25 Jalal Kazempour 11/26
Decomposable structures
Optimization problems with complicating variable(s)Example: a linear programming (LP) as the original problem
Subject to
is a complicating variable, i.e., if it is fixed to a given value ( ), then the originalproblem decomposes to 3 smaller optimization problems (subproblems)!
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26 June 2015DTU Electrical Engineering, Technical University of Denmark26 Jalal Kazempour 11/26
Decomposable structures
Optimization problems with complicating variable(s)Example: a linear programming (LP) as the original problem
Subject to
is a complicating variable, i.e., if it is fixed to a given value ( ), then the originalproblem decomposes to 3 smaller optimization problems (subproblems)!
Fixed term, it can be removed from objective function
Constraints including only variables
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26 June 2015DTU Electrical Engineering, Technical University of Denmark27 Jalal Kazempour 12/26
Decomposable structures
Optimization problems with complicating variable(s)Example: a linear programming (LP) as the original problem
Subproblem 1:
Subject to
Constraints including only variables
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26 June 2015DTU Electrical Engineering, Technical University of Denmark28 Jalal Kazempour 13/26
Decomposable structures
Optimization problems with complicating variable(s)Example: a linear programming (LP) as the original problem
Subproblem 2:
Constraint including only variables
Subject to
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26 June 2015DTU Electrical Engineering, Technical University of Denmark29 Jalal Kazempour 14/26
Decomposable structures
Optimization problems with complicating variable(s)Example: a linear programming (LP) as the original problem
Subproblem 3:
Subject to
Constraints including only variables
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26 June 2015DTU Electrical Engineering, Technical University of Denmark30 Jalal Kazempour 15/26
Decomposable structures
Optimization problems with complicating variable(s)
Subject to
Complicating variable
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Exercise 1
Six examples are available in the papers on your table.Please check them in the next 15 minutes, and identifywhether they are decomposable optimization problems ornot. If so,
• how? Identify complicating variables/constraints!
• Number of complicating variables/constraints?
• Number of subproblems?
Then, check your results with your peer.
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Example 1
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Example 1
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Example 2
Single‐node single‐hour market‐clearing problem (total cost minimization) with 3conventional generators and a single inelastic load:
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26 June 2015DTU Electrical Engineering, Technical University of Denmark35 Jalal Kazempour 18/26
Example 2
Single‐node single‐hour market‐clearing problem (total cost minimization) with 3conventional generators and a single inelastic load:
Complicating constraint: if relaxed, then the original problem decomposes to 3
subproblems, one per generator
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Example 3
Single‐node single‐hour market‐clearing problem (social welfare maximization) with 3conventional generators and a single elastic load:
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26 June 2015DTU Electrical Engineering, Technical University of Denmark37 Jalal Kazempour 19/26
Example 3
Single‐node single‐hour market‐clearing problem (social welfare maximization) with 3conventional generators and a single elastic load:
Complicating constraint: if relaxed, then the original problem decomposes to 4
subproblems, one per agent (i.e., 3 generators and 1 demand)
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Example 4
Single‐node multi‐hour (index: h) market‐clearing problem (social welfaremaximization) with 3 conventional generators and a single elastic load:
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26 June 2015DTU Electrical Engineering, Technical University of Denmark39 Jalal Kazempour 20/26
Example 4
Single‐node multi‐hour (index: h) market‐clearing problem (social welfaremaximization) with 3 conventional generators and a single elastic load:
Complicating constraints: if relaxed, then the original
problem decomposes to a set of subproblems, one per
agent per hour
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26 June 2015DTU Electrical Engineering, Technical University of Denmark40 Jalal Kazempour 20/26
Example 4
Single‐node multi‐hour (index: h) market‐clearing problem (social welfaremaximization) with 3 conventional generators and a single elastic load:
• Number of complicating constraints: ?• Number of subproblem: ?
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26 June 2015DTU Electrical Engineering, Technical University of Denmark41 Jalal Kazempour 20/26
Example 4
Single‐node multi‐hour (index: h) market‐clearing problem (social welfaremaximization) with 3 conventional generators and a single elastic load:
• Number of complicating constraints:|h|• Number of subproblem: 4|h|
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Example 5
Single‐node multi‐hour (index: h) market‐clearing problem (social welfaremaximization) with 3 conventional generators and a single elastic load (includingramping constraints):
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26 June 2015DTU Electrical Engineering, Technical University of Denmark43 Jalal Kazempour 21/26
Example 5
Single‐node multi‐hour (index: h) market‐clearing problem (social welfaremaximization) with 3 conventional generators and a single elastic load (includingramping constraints):
Investigate the following three options:1) relax ramping constraints only,2) relax balance constraints only,3) relax both.
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26 June 2015DTU Electrical Engineering, Technical University of Denmark44 Jalal Kazempour 21/26
Example 5
Single‐node multi‐hour (index: h) market‐clearing problem (social welfaremaximization) with 3 conventional generators and a single elastic load (includingramping constraints):
Option 1:• Number of complicating constraints: ?• Number of subproblem: ?
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Example 5
Single‐node multi‐hour (index: h) market‐clearing problem (social welfaremaximization) with 3 conventional generators and a single elastic load (includingramping constraints):
Option 1:• Number of complicating constraints:|h|• Number of subproblem: |h|
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26 June 2015DTU Electrical Engineering, Technical University of Denmark46 Jalal Kazempour 21/26
Example 5
Single‐node multi‐hour (index: h) market‐clearing problem (social welfaremaximization) with 3 conventional generators and a single elastic load (includingramping constraints):
Option 2:• Number of complicating constraints:?• Number of subproblem: ?
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Example 5
Single‐node multi‐hour (index: h) market‐clearing problem (social welfaremaximization) with 3 conventional generators and a single elastic load (includingramping constraints):
Option 2:• Number of complicating constraints:|h|• Number of subproblem: 3|h|+1
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Example 5
Single‐node multi‐hour (index: h) market‐clearing problem (social welfaremaximization) with 3 conventional generators and a single elastic load (includingramping constraints):
Option 3:• Number of complicating constraints: ?• Number of subproblem: ?
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Example 5
Single‐node multi‐hour (index: h) market‐clearing problem (social welfaremaximization) with 3 conventional generators and a single elastic load (includingramping constraints):
Option 3:• Number of complicating constraints: 2|h|• Number of subproblem: 4|h|
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Example 6
Single‐node single‐year (static) generation expansion problem, considering two existinggenerators (units 1 and 2) and one candidate generator (unit 3)
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Example 6
Single‐node single‐year (static) generation expansion problem, considering two existinggenerators (units 1 and 2) and one candidate generator (unit 3)
Expansion cost of candidate unit Operational cost
of (existing and candidate) units
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Example 6
Single‐node single‐year (static) generation expansion problem, considering two existinggenerators (units 1 and 2) and one candidate generator (unit 3)
Investigate the following two options:1) x3 is a complicating variable,2) Balance conditions are complicating
constraints.
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Example 6
Single‐node single‐year (static) generation expansion problem, considering two existinggenerators (units 1 and 2) and one candidate generator (unit 3)
Option 1 (fixing x3):• Number of complicating variables: ?• Number of subproblem: ?
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Example 6
Single‐node single‐year (static) generation expansion problem, considering two existinggenerators (units 1 and 2) and one candidate generator (unit 3)
Option 1 (fixing x3):• Number of complicating variables: 1• Number of subproblem: |h|
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Example 6
Single‐node single‐year (static) generation expansion problem, considering two existinggenerators (units 1 and 2) and one candidate generator (unit 3)
Option 2 (relaxing balance constraints):• Number of complicating constraints: ?• Number of subproblem: ?
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Example 6
Single‐node single‐year (static) generation expansion problem, considering two existinggenerators (units 1 and 2) and one candidate generator (unit 3)
Option 2 (relaxing balance constraints):• Number of complicating constraints: |h|• Number of subproblem: 2|h|+1
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Optional Assignment 1
Consider the following generation expansion problem, which is decomposable byfixing complicating variable x3. Derive its corresponding dual optimization, andcheck whether it is decomposable too. Any general conclusion?
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Optional Assignment 2
Consider the following compact form of stochastic market‐clearing problem. List alloptions for decomposing this problem, and indicate the complicatingvariables/constraints. The ideal options decompose the original problem byscenario (i.e., one subproblem per scenario), since this problem may include a largenumber of scenarios.
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Optional Assignment 3
Think of any other optimization problem in energy systems – it could be yourcurrent or previous (or future!) research work. Is it decomposable? Write it in acompact form (similar to optional assignment 2) and then discuss how it isdecomposable.
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26 June 2015DTU Electrical Engineering, Technical University of Denmark
Thanks for your attention!
Email: [email protected]
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