Network flow problem Flow conservation constraints � � q k − q k = c i k ∈ A ini k ∈ A fin i i ∀ i ∈ N C demand nodes � � 0 ≤ q k − q k ≤ s i k ∈ A ini k ∈ A fin i i ∀ i ∈ N S supply nodes Box Constraints q k ≤ q k ≤ ¯ q k ¯ ∀ k ∈ A flow bounds 2 ≤ ¯ p 2 p 2 i ≤ p i i ¯ ∀ i ∈ N pressure bounds
Network flow problem Flow conservation constraints � � q k − q k = c i k ∈ A ini k ∈ A fin i i ∀ i ∈ N C demand nodes � � 0 ≤ q k − q k ≤ s i k ∈ A ini k ∈ A fin i i ∀ i ∈ N S supply nodes Box Constraints q k ≤ q k ≤ ¯ q k ¯ ∀ k ∈ A flow bounds Variables of the optimization problem 2 ≤ ¯ p 2 p 2 i ≤ p i Flow through each pipe i ¯ ∀ i ∈ N pressure bounds Pressure at each node
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Gass loss equations Given a pipe between two nodes i and j , we have p i 2 + p j 2 2 = 16 L k λ k Z ( p m , T m ) RT m | q ij | q ij + 2 g 2 − p j � � p i h j − h i π 2 D 5 2 Z ( p m , T m ) RT m k A two-step SLP for MINLP problems RSME 2017 5/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Gass loss equations Given a pipe between two nodes i and j , we have p i 2 + p j 2 2 = 16 L k λ k Z ( p m , T m ) RT m | q ij | q ij + 2 g 2 − p j � � p i h j − h i π 2 D 5 2 Z ( p m , T m ) RT m k A two-step SLP for MINLP problems RSME 2017 5/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Gass loss equations Given a pipe between two nodes i and j , we have p i 2 + p j 2 2 = 16 L k λ k Z ( p m , T m ) RT m | q ij | q ij + 2 g 2 − p j � � p i h j − h i π 2 D 5 2 Z ( p m , T m ) RT m k A two-step SLP for MINLP problems RSME 2017 5/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Gass loss equations Given a pipe between two nodes i and j , we have p i 2 + p j 2 2 = 16 L k λ k Z ( p m , T m ) RT m | q ij | q ij + 2 g 2 − p j � � p i h j − h i π 2 D 5 2 Z ( p m , T m ) RT m k As many nonlinear constraints as pipes A two-step SLP for MINLP problems RSME 2017 5/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Gas consumption at compressors Given input pressure p i and output pressure p j , we have � � 1 γ ( p j γ − 1 γ − 1 g ij = γ − 1 Z ( p m , T in ) RT in ) q ij e h H c p i A two-step SLP for MINLP problems RSME 2017 6/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Gas consumption at compressors Given input pressure p i and output pressure p j , we have � � 1 γ ( p j γ − 1 γ − 1 g ij = γ − 1 Z ( p m , T in ) RT in ) q ij e h H c p i As many nonlinear constraints as compressors in the network A two-step SLP for MINLP problems RSME 2017 6/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Nonlinear nonconvex optimization problem A two-step SLP for MINLP problems RSME 2017 7/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Nonlinear nonconvex optimization problem Obj. Function: min � k ∈ A c g k A two-step SLP for MINLP problems RSME 2017 7/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Nonlinear nonconvex optimization problem Obj. Function: min � k ∈ A c g k Box Constraints 2 ≤ ¯ p 2 p 2 i ≤ p i ∀ i ∈ N pressure bounds i ¯ q k ≤ q k ≤ ¯ q k ∀ k ∈ A flow bounds ¯ Flow conservation constraints � � ∀ i ∈ N C flow conservation at demand nodes q k − q k = c i k ∈ A ini k ∈ A fin i i � � ∀ i ∈ N S flow conservation at supply nodes 0 ≤ q k − q k ≤ s i k ∈ A ini k ∈ A fin i i Gas loss constraints 2 = 16 L k λ k 2 − p j ∀ k ∈ A n gas loss ( λ k Weymouth) p i Z ( p m , T m ) RT m | q k | q k + π 2 D 5 k 2 + p j 2 + 2 g p i � � h j − h i height difference term RT m 2 Z ( p m , T m ) Gas consumption constraints 1 � � γ ( p j γ − 1 g k = γ − 1 Z ( p m , T in ) RT in p i ) − 1 q k γ e h H c A two-step SLP for MINLP problems RSME 2017 7/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Nonlinear nonconvex optimization problem (continuous) Obj. Function: min � k ∈ A c g k Box Constraints 2 ≤ ¯ p 2 p 2 i ≤ p i ∀ i ∈ N pressure bounds i ¯ q k ≤ q k ≤ ¯ q k ∀ k ∈ A flow bounds ¯ Flow conservation constraints � � ∀ i ∈ N C flow conservation at demand nodes q k − q k = c i k ∈ A ini k ∈ A fin i i � � ∀ i ∈ N S flow conservation at supply nodes 0 ≤ q k − q k ≤ s i k ∈ A ini k ∈ A fin i i Gas loss constraints 2 = 16 L k λ k 2 − p j ∀ k ∈ A n gas loss ( λ k Weymouth) p i Z ( p m , T m ) RT m | q k | q k + π 2 D 5 k 2 + p j 2 + 2 g p i � � h j − h i height difference term RT m 2 Z ( p m , T m ) Gas consumption constraints 1 � � γ ( p j γ − 1 g k = γ − 1 Z ( p m , T in ) RT in p i ) − 1 q k γ e h H c A two-step SLP for MINLP problems RSME 2017 7/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Size and complexity of real instances A two-step SLP for MINLP problems RSME 2017 8/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Size and complexity of real instances Spanish primary gas network ≈ 1000 variables ( ≈ 500 pipes and ≈ 500 nodes) ≈ 1000 constraints (and ≈ 2000 box constraints) ≈ 500 constraints are nonlinear A two-step SLP for MINLP problems RSME 2017 8/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Size and complexity of real instances Spanish primary gas network ≈ 1000 variables ( ≈ 500 pipes and ≈ 500 nodes) ≈ 1000 constraints (and ≈ 2000 box constraints) ≈ 500 constraints are nonlinear To be solved routinely by the company A two-step SLP for MINLP problems RSME 2017 8/25
(A twist on) Sequential Linear Programming Algorithms Optimization in Gas Transmission Networks 1 (A twist on) Sequential Linear Programming Algorithms 2 Numerical Results 3
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Approaches to solve the problem Spanish primary gas network ≈ 1000 variables ( ≈ 500 pipes and ≈ 500 nodes) ≈ 1000 constraints (and ≈ 2000 box constraints) ≈ 500 constraints are nonlinear A two-step SLP for MINLP problems RSME 2017 9/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Approaches to solve the problem Spanish primary gas network ≈ 1000 variables ( ≈ 500 pipes and ≈ 500 nodes) ≈ 1000 constraints (and ≈ 2000 box constraints) ≈ 500 constraints are nonlinear How to solve this problem? A two-step SLP for MINLP problems RSME 2017 9/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Approaches to solve the problem Spanish primary gas network ≈ 1000 variables ( ≈ 500 pipes and ≈ 500 nodes) ≈ 1000 constraints (and ≈ 2000 box constraints) ≈ 500 constraints are nonlinear How to solve this problem? Global optimization algorithms on approximations of the problem nonlinearities = ⇒ piecewise linear functions + integer variables A two-step SLP for MINLP problems RSME 2017 9/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Approaches to solve the problem Spanish primary gas network ≈ 1000 variables ( ≈ 500 pipes and ≈ 500 nodes) ≈ 1000 constraints (and ≈ 2000 box constraints) ≈ 500 constraints are nonlinear How to solve this problem? Global optimization algorithms on approximations of the problem (cannot handle real-size problems) nonlinearities = ⇒ piecewise linear functions + integer variables A two-step SLP for MINLP problems RSME 2017 9/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Approaches to solve the problem Spanish primary gas network ≈ 1000 variables ( ≈ 500 pipes and ≈ 500 nodes) ≈ 1000 constraints (and ≈ 2000 box constraints) ≈ 500 constraints are nonlinear How to solve this problem? Global optimization algorithms on approximations of the problem (cannot handle real-size problems) nonlinearities = ⇒ piecewise linear functions + integer variables Local optimization algorithms such as sequential linear programming, SLP , or sequential quadratic programming, SQP A two-step SLP for MINLP problems RSME 2017 9/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Our initial approach Classic SLP A two-step SLP for MINLP problems RSME 2017 10/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Our initial approach Classic SLP We get a solution using Classic SLP A two-step SLP for MINLP problems RSME 2017 10/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Our initial approach Classic SLP + Control Theory We get a solution using Classic SLP We refine it using control theory by including some second order elements A two-step SLP for MINLP problems RSME 2017 10/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Our initial approach Classic SLP + Control Theory We get a solution using Classic SLP We refine it using control theory by including some second order elements Nothing specially original so far A two-step SLP for MINLP problems RSME 2017 10/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Our initial approach Classic SLP We get a solution using Classic SLP Nothing specially original so far A two-step SLP for MINLP problems RSME 2017 10/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Additional network elements Elements that require the use of binary variables A two-step SLP for MINLP problems RSME 2017 11/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Additional network elements Elements that require the use of binary variables Different types of control valves Operational ranges of each compressor station Boil-off gas at regasification plants A two-step SLP for MINLP problems RSME 2017 11/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Additional network elements Elements that require the use of binary variables Different types of control valves Operational ranges of each compressor station Boil-off gas at regasification plants Mixed-integer nonlinear nonconvex programming problem ≈ 1000 continuous variables and 1000 constraints No more than 100-200 binary variables A two-step SLP for MINLP problems RSME 2017 11/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Additional network elements Elements that require the use of binary variables Different types of control valves Operational ranges of each compressor station Boil-off gas at regasification plants Mixed-integer nonlinear nonconvex programming problem ≈ 1000 continuous variables and 1000 constraints No more than 100-200 binary variables How are these problems normally tackled? A two-step SLP for MINLP problems RSME 2017 11/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Additional network elements Elements that require the use of binary variables Different types of control valves Operational ranges of each compressor station Boil-off gas at regasification plants Mixed-integer nonlinear nonconvex programming problem ≈ 1000 continuous variables and 1000 constraints No more than 100-200 binary variables How are these problems normally tackled? Two-step algorithms A two-step SLP for MINLP problems RSME 2017 11/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Additional network elements Elements that require the use of binary variables Different types of control valves Operational ranges of each compressor station Boil-off gas at regasification plants Mixed-integer nonlinear nonconvex programming problem ≈ 1000 continuous variables and 1000 constraints No more than 100-200 binary variables How are these problems normally tackled? Two-step algorithms Step 1. Study a simplified version of the problem to fix all binary choices A two-step SLP for MINLP problems RSME 2017 11/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Additional network elements Elements that require the use of binary variables Different types of control valves Operational ranges of each compressor station Boil-off gas at regasification plants Mixed-integer nonlinear nonconvex programming problem ≈ 1000 continuous variables and 1000 constraints No more than 100-200 binary variables How are these problems normally tackled? Two-step algorithms Step 1. Study a simplified version of the problem to fix all binary choices Step 2. Apply SLP , SQP ,. . . to the resulting continuous problem A two-step SLP for MINLP problems RSME 2017 11/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Our two-step approach for MINLP problems Classic SLP We get a solution using Classic SLP A two-step SLP for MINLP problems RSME 2017 12/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Our two-step approach for MINLP problems Classic SLP Step 2. Classic SLP. Binary variables already fixed We get a solution using Classic SLP A two-step SLP for MINLP problems RSME 2017 12/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Our two-step approach for MINLP problems Classic SLP Step 1. Step 2. Classic SLP. Binary variables already fixed We get a solution using Classic SLP A two-step SLP for MINLP problems RSME 2017 12/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Our two-step approach for MINLP problems 2SLP : SLP-NTR + Classic SLP Step 1. SLP-NTR ( N o T rust R egion) Step 2. Classic SLP. Binary variables already fixed We get a solution using Classic SLP A two-step SLP for MINLP problems RSME 2017 12/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Our two-step approach for MINLP problems 2SLP : SLP-NTR + Classic SLP Step 1. SLP-NTR ( N o T rust R egion) The solution of this step is used to fix the binary variables Step 2. Classic SLP. Binary variables already fixed We get a solution using Classic SLP A two-step SLP for MINLP problems RSME 2017 12/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Our two-step approach for MINLP problems 2SLP : SLP-NTR + Classic SLP Step 1. SLP-NTR ( N o T rust R egion) The solution of this step is used to fix the binary variables Step 2. Classic SLP. Binary variables already fixed We get a solution using Classic SLP Step 1 runs on the full model. No simplification needed A two-step SLP for MINLP problems RSME 2017 12/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results SLP-NTR ( N o T rust R egion) Nonlinear programming problem: NLP minimize f ( x ) subject to inequality contraints g i ( x ) ≤ 0 , i = 1 , · · · , m h j ( x ) = 0 , j = 1 , · · · , l equality constrains x ∈ X = { x ∈ R n : A x ≤ b } linear constraints where f , g i and h j are nonlinear functions. A two-step SLP for MINLP problems RSME 2017 13/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results SLP-NTR ( N o T rust R egion) Classic SLP A two-step SLP for MINLP problems RSME 2017 14/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results SLP-NTR ( N o T rust R egion) Classic SLP At iteration k we have a candidate solution x k A two-step SLP for MINLP problems RSME 2017 14/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results SLP-NTR ( N o T rust R egion) Classic SLP At iteration k we have a candidate solution x k We solve the linearization of NLP about x k , LP( x k ): A two-step SLP for MINLP problems RSME 2017 14/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results SLP-NTR ( N o T rust R egion) Classic SLP At iteration k we have a candidate solution x k We solve the linearization of NLP about x k , LP( x k ): ∇ f ( x k ) t x minimize subject to g i ( x k ) + ∇ g i ( x k ) t ( x − x k ) ≤ 0 i = 1 , · · · , m inequality constraints h j ( x k ) + ∇ h j ( x k ) t ( x − x k ) = 0 equality constraints j = 1 , · · · , l x ∈ X = { x ∈ R n : A x ≤ b } linear constraints − d k ≤ x − x k ≤ d k trust region A two-step SLP for MINLP problems RSME 2017 14/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results SLP-NTR ( N o T rust R egion) Classic SLP At iteration k we have a candidate solution x k We solve the linearization of NLP about x k , LP( x k ): ∇ f ( x k ) t x minimize subject to g i ( x k ) + ∇ g i ( x k ) t ( x − x k ) ≤ 0 i = 1 , · · · , m inequality constraints h j ( x k ) + ∇ h j ( x k ) t ( x − x k ) = 0 equality constraints j = 1 , · · · , l x ∈ X = { x ∈ R n : A x ≤ b } linear constraints − d k ≤ x − x k ≤ d k trust region A two-step SLP for MINLP problems RSME 2017 14/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results SLP-NTR ( N o T rust R egion) Classic SLP At iteration k we have a candidate solution x k We solve the linearization of NLP about x k , LP( x k ): ∇ f ( x k ) t x minimize subject to g i ( x k ) + ∇ g i ( x k ) t ( x − x k ) ≤ 0 i = 1 , · · · , m inequality constraints h j ( x k ) + ∇ h j ( x k ) t ( x − x k ) = 0 equality constraints j = 1 , · · · , l x ∈ X = { x ∈ R n : A x ≤ b } linear constraints − d k ≤ x − x k ≤ d k trust region Hard to accommodate binary variables with the trust region A two-step SLP for MINLP problems RSME 2017 14/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results SLP-NTR ( N o T rust R egion) SLP-NTR At iteration k we have a candidate solution x k We solve the linearization of NLP about x k , LP( x k ): ∇ f ( x k ) t x minimize subject to g i ( x k ) + ∇ g i ( x k ) t ( x − x k ) ≤ 0 inequality constraints i = 1 , · · · , m h j ( x k ) + ∇ h j ( x k ) t ( x − x k ) = 0 j = 1 , · · · , l equality constraints x ∈ X = { x ∈ R n : A x ≤ b } linear constraints − d k ≤ x − x k ≤ d k trust region A two-step SLP for MINLP problems RSME 2017 14/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results SLP-NTR ( N o T rust R egion) SLP-NTR At iteration k we have a candidate solution x k We solve the linearization of NLP about x k , LP( x k ): ∇ f ( x k ) t x minimize subject to g i ( x k ) + ∇ g i ( x k ) t ( x − x k ) ≤ 0 inequality constraints i = 1 , · · · , m h j ( x k ) + ∇ h j ( x k ) t ( x − x k ) = 0 j = 1 , · · · , l equality constraints x ∈ X = { x ∈ R n : A x ≤ b } linear constraints − d k ≤ x − x k ≤ d k trust region / / / / / / / / / / / / / / / / / / / / / / / / We remove the constraints that define the trust region A two-step SLP for MINLP problems RSME 2017 14/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results SLP-NTR ( N o T rust R egion) SLP-NTR At iteration k we have a candidate solution x k We solve the linearization of NLP about x k , LP( x k ): ∇ f ( x k ) t x minimize subject to g i ( x k ) + ∇ g i ( x k ) t ( x − x k ) ≤ 0 inequality constraints i = 1 , · · · , m h j ( x k ) + ∇ h j ( x k ) t ( x − x k ) = 0 j = 1 , · · · , l equality constraints x ∈ X = { x ∈ R n : A x ≤ b } linear constraints − d k ≤ x − x k ≤ d k trust region / / / / / / / / / / / / / / / / / / / / / / / / We remove the constraints that define the trust region Straightforward inclusion of binary variables A two-step SLP for MINLP problems RSME 2017 14/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results SLP-NTR ( N o T rust R egion) SLP-NTR At iteration k we have a candidate solution x k We solve the linearization of NLP about x k , LP( x k ): ∇ f ( x k ) t x minimize subject to g i ( x k ) + ∇ g i ( x k ) t ( x − x k ) ≤ 0 inequality constraints i = 1 , · · · , m h j ( x k ) + ∇ h j ( x k ) t ( x − x k ) = 0 j = 1 , · · · , l equality constraints x ∈ X = { x ∈ R n : A x ≤ b } linear constraints − d k ≤ x − x k ≤ d k trust region / / / / / / / / / / / / / / / / / / / / / / / / We remove the constraints that define the trust region Straightforward inclusion of binary variables Theoretical justification for the removal of the trust region? A two-step SLP for MINLP problems RSME 2017 14/25
SLP-NTR vs classic SLP (in the continuous case, NLP problems)
SLP-NTR vs classic SLP (in the continuous case, NLP problems) Classic SLP ++ Accumulation points of the sequence are KKT points of NLP ++ In practice it normally converges
SLP-NTR vs classic SLP (in the continuous case, NLP problems) Classic SLP ++ Accumulation points of the sequence are KKT points of NLP ++ In practice it normally converges −− A number of parameters have to be tuned −− Hard to accommodate binary variables
SLP-NTR vs classic SLP (in the continuous case, NLP problems) Classic SLP ++ Accumulation points of the sequence are KKT points of NLP ++ In practice it normally converges −− A number of parameters have to be tuned −− Hard to accommodate binary variables SLP-NTR ( N o T rust R egion)
SLP-NTR vs classic SLP (in the continuous case, NLP problems) Classic SLP ++ Accumulation points of the sequence are KKT points of NLP ++ In practice it normally converges −− A number of parameters have to be tuned −− Hard to accommodate binary variables SLP-NTR ( N o T rust R egion) ++ If the sequence converges, the limit is a KKT point of NLP
SLP-NTR vs classic SLP (in the continuous case, NLP problems) Classic SLP ++ Accumulation points of the sequence are KKT points of NLP ++ In practice it normally converges −− A number of parameters have to be tuned −− Hard to accommodate binary variables SLP-NTR ( N o T rust R egion) ++ If the sequence converges, the limit is a KKT point of NLP −− Other accumulation points may not be KKT points of NLP
SLP-NTR vs classic SLP (in the continuous case, NLP problems) Classic SLP ++ Accumulation points of the sequence are KKT points of NLP ++ In practice it normally converges −− A number of parameters have to be tuned −− Hard to accommodate binary variables SLP-NTR ( N o T rust R egion) ++ If the sequence converges, the limit is a KKT point of NLP −− Other accumulation points may not be KKT points of NLP −− It cannot converge to interior points of the feasible set
SLP-NTR vs classic SLP (in the continuous case, NLP problems) Classic SLP ++ Accumulation points of the sequence are KKT points of NLP ++ In practice it normally converges −− A number of parameters have to be tuned −− Hard to accommodate binary variables SLP-NTR ( N o T rust R egion) ++ If the sequence converges, the limit is a KKT point of NLP −− Other accumulation points may not be KKT points of NLP −− It cannot converge to interior points of the feasible set ( min x ∈ [ − 1 , 1] x 2 )
SLP-NTR vs classic SLP (in the continuous case, NLP problems) Classic SLP ++ Accumulation points of the sequence are KKT points of NLP ++ In practice it normally converges −− A number of parameters have to be tuned −− Hard to accommodate binary variables SLP-NTR ( N o T rust R egion) ++ If the sequence converges, the limit is a KKT point of NLP −− Other accumulation points may not be KKT points of NLP −− It cannot converge to interior points of the feasible set ( min x ∈ [ − 1 , 1] x 2 ) (Not so critical, since we run 2SLP: SLP-NTR+CSLP)
SLP-NTR vs classic SLP (in the continuous case, NLP problems) Classic SLP ++ Accumulation points of the sequence are KKT points of NLP ++ In practice it normally converges −− A number of parameters have to be tuned −− Hard to accommodate binary variables SLP-NTR ( N o T rust R egion) ++ If the sequence converges, the limit is a KKT point of NLP −− Other accumulation points may not be KKT points of NLP −− It cannot converge to interior points of the feasible set ( min x ∈ [ − 1 , 1] x 2 ) (Not so critical, since we run 2SLP: SLP-NTR+CSLP) −− Less stable in terms of convergence ( e.g. , cycling )
SLP-NTR vs classic SLP (in the continuous case, NLP problems) Classic SLP ++ Accumulation points of the sequence are KKT points of NLP ++ In practice it normally converges −− A number of parameters have to be tuned −− Hard to accommodate binary variables SLP-NTR ( N o T rust R egion) ++ If the sequence converges, the limit is a KKT point of NLP −− Other accumulation points may not be KKT points of NLP −− It cannot converge to interior points of the feasible set ( min x ∈ [ − 1 , 1] x 2 ) (Not so critical, since we run 2SLP: SLP-NTR+CSLP) −− Less stable in terms of convergence ( e.g. , cycling ) ++ If two consecutive points of { x k } are sufficiently close − → almost KKT of NLP
SLP-NTR vs classic SLP (in the continuous case, NLP problems) Classic SLP ++ Accumulation points of the sequence are KKT points of NLP ++ In practice it normally converges −− A number of parameters have to be tuned −− Hard to accommodate binary variables SLP-NTR ( N o T rust R egion) ++ If the sequence converges, the limit is a KKT point of NLP −− Other accumulation points may not be KKT points of NLP −− It cannot converge to interior points of the feasible set ( min x ∈ [ − 1 , 1] x 2 ) (Not so critical, since we run 2SLP: SLP-NTR+CSLP) −− Less stable in terms of convergence ( e.g. , cycling ) ++ If two consecutive points of { x k } are sufficiently close − → almost KKT of NLP
SLP-NTR vs classic SLP (in the continuous case, NLP problems) Classic SLP ++ Accumulation points of the sequence are KKT points of NLP ++ In practice it normally converges −− A number of parameters have to be tuned −− Hard to accommodate binary variables SLP-NTR ( N o T rust R egion) ++ If the sequence converges, the limit is a KKT point of NLP −− Other accumulation points may not be KKT points of NLP −− It cannot converge to interior points of the feasible set ( min x ∈ [ − 1 , 1] x 2 ) (Not so critical, since we run 2SLP: SLP-NTR+CSLP) −− Less stable in terms of convergence ( e.g. , cycling ) ++ If two consecutive points of { x k } are sufficiently close − → almost KKT of NLP ++ Very easy to implement. No parameters to be tuned
SLP-NTR vs classic SLP (in the continuous case, NLP problems) Classic SLP ++ Accumulation points of the sequence are KKT points of NLP ++ In practice it normally converges −− A number of parameters have to be tuned −− Hard to accommodate binary variables SLP-NTR ( N o T rust R egion) ++ If the sequence converges, the limit is a KKT point of NLP −− Other accumulation points may not be KKT points of NLP −− It cannot converge to interior points of the feasible set ( min x ∈ [ − 1 , 1] x 2 ) (Not so critical, since we run 2SLP: SLP-NTR+CSLP) −− Less stable in terms of convergence ( e.g. , cycling ) ++ If two consecutive points of { x k } are sufficiently close − → almost KKT of NLP ++ Very easy to implement. No parameters to be tuned ++ It is straightforward to incorporate binary variables
SLP-NTR vs classic SLP (in the continuous case, NLP problems) Classic SLP ++ Accumulation points of the sequence are KKT points of NLP ++ In practice it normally converges −− A number of parameters have to be tuned −− Hard to accommodate binary variables SLP-NTR ( N o T rust R egion) ++ If the sequence converges, the limit is a KKT point of NLP −− Other accumulation points may not be KKT points of NLP −− It cannot converge to interior points of the feasible set ( min x ∈ [ − 1 , 1] x 2 ) (Not so critical, since we run 2SLP: SLP-NTR+CSLP) −− Less stable in terms of convergence ( e.g. , cycling ) ++ If two consecutive points of { x k } are sufficiently close − → almost KKT of NLP ++ Very easy to implement. No parameters to be tuned ++ It is straightforward to incorporate binary variables ++ SLP-NTR competitive with classic SLP for gas network problems and multicommodity flow problems
SLP-NTR vs classic SLP (in the continuous case, NLP problems) Classic SLP ++ Accumulation points of the sequence are KKT points of NLP ++ In practice it normally converges −− A number of parameters have to be tuned −− Hard to accommodate binary variables SLP-NTR ( N o T rust R egion) ++ If the sequence converges, the limit is a KKT point of NLP −− Other accumulation points may not be KKT points of NLP −− It cannot converge to interior points of the feasible set ( min x ∈ [ − 1 , 1] x 2 ) (Not so critical, since we run 2SLP: SLP-NTR+CSLP) −− Less stable in terms of convergence ( e.g. , cycling ) ++ If two consecutive points of { x k } are sufficiently close − → almost KKT of NLP ++ Very easy to implement. No parameters to be tuned ++ It is straightforward to incorporate binary variables ++ SLP-NTR competitive with classic SLP for gas network problems and multicommodity flow problems
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Summary (algorithm for MINLP problems) 2SLP : SLP-NTR + Classic SLP Step 1. SLP-NTR ( N o T rust R egion) Step 2. Classic SLP A two-step SLP for MINLP problems RSME 2017 16/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Summary (algorithm for MINLP problems) 2SLP : SLP-NTR + Classic SLP Step 1. SLP-NTR ( N o T rust R egion) Step 2. Classic SLP Features of our two-step approach A two-step SLP for MINLP problems RSME 2017 16/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Summary (algorithm for MINLP problems) 2SLP : SLP-NTR + Classic SLP Step 1. SLP-NTR ( N o T rust R egion) Step 2. Classic SLP Features of our two-step approach Easy to implement A two-step SLP for MINLP problems RSME 2017 16/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Summary (algorithm for MINLP problems) 2SLP : SLP-NTR + Classic SLP Step 1. SLP-NTR ( N o T rust R egion) Step 2. Classic SLP Features of our two-step approach Easy to implement Step 1 runs on the full model . No simplification needed A two-step SLP for MINLP problems RSME 2017 16/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Summary (algorithm for MINLP problems) 2SLP : SLP-NTR + Classic SLP Step 1. SLP-NTR ( N o T rust R egion) Step 2. Classic SLP Features of our two-step approach Easy to implement Step 1 runs on the full model . No simplification needed Step 2 “guarantees” convergence A two-step SLP for MINLP problems RSME 2017 16/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Summary (algorithm for MINLP problems) 2SLP : SLP-NTR + Classic SLP Step 1. SLP-NTR ( N o T rust R egion) Step 2. Classic SLP Features of our two-step approach Easy to implement Step 1 runs on the full model . No simplification needed Step 2 “guarantees” convergence Good practical behavior ( < 5 minutes running time on Spanish network) Significant cost reduction with respect to operation schemes reported by the Transmission System Operator (whose software does not optimize) A two-step SLP for MINLP problems RSME 2017 16/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Summary (algorithm for MINLP problems) 2SLP : SLP-NTR + Classic SLP Step 1. SLP-NTR ( N o T rust R egion) Step 2. Classic SLP Features of our two-step approach Easy to implement Step 1 runs on the full model . No simplification needed Step 2 “guarantees” convergence Good practical behavior ( < 5 minutes running time on Spanish network) Significant cost reduction with respect to operation schemes reported by the Transmission System Operator (whose software does not optimize) Limitation: No bounds/gap to optimality A two-step SLP for MINLP problems RSME 2017 16/25
Optimization in Gas Transmission Networks (A twist on) SLP Algorithms Numerical Results Our contribution A two-step SLP for MINLP problems RSME 2017 17/25
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