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Distributed Optimization
Lecture 1: Optimization problems with decomposable structure
Jalal Kazempour
26 June 2015June 21, 2019
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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Decomposition: main idea
Decomposition
Original (non‐decomposed) optimization problem with decomposable structure
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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)
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
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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Decomposable structure
Optimization problems with decomposable structure:
Problems with complicating variable(s):
Problems with complicating constraint(s):
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)!
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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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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Decomposable structures
Optimization problems with complicating constraint(s)Example: a linear programming (LP) as the original problem
Subject to
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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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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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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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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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Decomposable structures
Optimization problems with complicating constraint(s)
Subject to
Complicating constraint
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Decomposable structures
Optimization problems with complicating variable(s)Example: a linear programming (LP) as the original problem
Subject to
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Decomposable structures
Optimization problems with complicating variable(s)Example: a linear programming (LP) as the original problem
Subject to
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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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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
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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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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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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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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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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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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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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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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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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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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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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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.
26 June 2015DTU Electrical Engineering, Technical University of Denmark
Thanks for your attention!
Email: [email protected]
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