Cubic programming and beyond?Reference request: how to model nonlinear regression?Single reference for Mixed Integer Programming formulations to linearize, handle logical constraints and disjunctive constraints, do Big M, etc?McCormick envelopes and nonlinear constraintsCPLEX non-convex Quadratic Programming algorithmsHeuristics for mixed integer linear and nonlinear programsSum of Max terms maximization
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Cubic programming and beyond?
Reference request: how to model nonlinear regression?Single reference for Mixed Integer Programming formulations to linearize, handle logical constraints and disjunctive constraints, do Big M, etc?McCormick envelopes and nonlinear constraintsCPLEX non-convex Quadratic Programming algorithmsHeuristics for mixed integer linear and nonlinear programsSum of Max terms maximization
$begingroup$
It is almost inevitable in Operations Research to come across linear or quadratic programming problems. The overall structures of these problems are below: beginalignbeginarrayll
sfLinear\
max & bf c^top x\
texts.t. & Abf xle b \
textand & bf x ge 0
endarrayquadquadquadquadbeginarrayllsfQuadratic\min ½bfx^topQbf x+c^top x\texts.t. & Abf xpreceq b\endarrayendalign Both types of programming have their own (if not overlapping) applications; see, for example McCarl et al. (1977).
However, I have rarely heard of specific names for higher-order programming problems other than the generic "non-linear programming" term.
How much work has gone into the study of cubic/quartic etc. programming? What do the structures of these problems look like, and are there any specific examples of where they can be useful?
Reference
[1] McCarl, B. A., Moskowitz, H., Furtan, H. (1977). Quadratic programming applications. Omega. 5(1):43-55.
reference-request nonlinear-programming
asked 8 hours ago
TheSimpliFireTheSimpliFire
1,0563 silver badges25 bronze badges
$endgroup$
add a comment |
$begingroup$
It is almost inevitable in Operations Research to come across linear or quadratic programming problems. The overall structures of these problems are below: beginalignbeginarrayll
sfLinear\
max & bf c^top x\
texts.t. & Abf xle b \
textand & bf x ge 0
endarrayquadquadquadquadbeginarrayllsfQuadratic\min ½bfx^topQbf x+c^top x\texts.t. & Abf xpreceq b\endarrayendalign Both types of programming have their own (if not overlapping) applications; see, for example McCarl et al. (1977).
However, I have rarely heard of specific names for higher-order programming problems other than the generic "non-linear programming" term.
How much work has gone into the study of cubic/quartic etc. programming? What do the structures of these problems look like, and are there any specific examples of where they can be useful?
Reference
[1] McCarl, B. A., Moskowitz, H., Furtan, H. (1977). Quadratic programming applications. Omega. 5(1):43-55.
reference-request nonlinear-programming
asked 8 hours ago
TheSimpliFireTheSimpliFire
1,0563 silver badges25 bronze badges
$endgroup$
add a comment |
$begingroup$
It is almost inevitable in Operations Research to come across linear or quadratic programming problems. The overall structures of these problems are below: beginalignbeginarrayll
sfLinear\
max & bf c^top x\
texts.t. & Abf xle b \
textand & bf x ge 0
endarrayquadquadquadquadbeginarrayllsfQuadratic\min ½bfx^topQbf x+c^top x\texts.t. & Abf xpreceq b\endarrayendalign Both types of programming have their own (if not overlapping) applications; see, for example McCarl et al. (1977).
However, I have rarely heard of specific names for higher-order programming problems other than the generic "non-linear programming" term.
How much work has gone into the study of cubic/quartic etc. programming? What do the structures of these problems look like, and are there any specific examples of where they can be useful?
Reference
[1] McCarl, B. A., Moskowitz, H., Furtan, H. (1977). Quadratic programming applications. Omega. 5(1):43-55.
reference-request nonlinear-programming
asked 8 hours ago
TheSimpliFireTheSimpliFire
1,0563 silver badges25 bronze badges
$endgroup$
It is almost inevitable in Operations Research to come across linear or quadratic programming problems. The overall structures of these problems are below: beginalignbeginarrayll
sfLinear\
max & bf c^top x\
texts.t. & Abf xle b \
textand & bf x ge 0
endarrayquadquadquadquadbeginarrayllsfQuadratic\min ½bfx^topQbf x+c^top x\texts.t. & Abf xpreceq b\endarrayendalign Both types of programming have their own (if not overlapping) applications; see, for example McCarl et al. (1977).
However, I have rarely heard of specific names for higher-order programming problems other than the generic "non-linear programming" term.
How much work has gone into the study of cubic/quartic etc. programming? What do the structures of these problems look like, and are there any specific examples of where they can be useful?
Reference
[1] McCarl, B. A., Moskowitz, H., Furtan, H. (1977). Quadratic programming applications. Omega. 5(1):43-55.
reference-request nonlinear-programming
reference-request nonlinear-programming
asked 8 hours ago
TheSimpliFireTheSimpliFire
1,0563 silver badges25 bronze badges
asked 8 hours ago
TheSimpliFireTheSimpliFire
1,0563 silver badges25 bronze badges
asked 8 hours ago
TheSimpliFireTheSimpliFire
1,0563 silver badges25 bronze badges
asked 8 hours ago
TheSimpliFireTheSimpliFire
1,0563 silver badges25 bronze badges
asked 8 hours ago
TheSimpliFireTheSimpliFire
1,0563 silver badges25 bronze badges
1,0563 silver badges25 bronze badges
add a comment |
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
I am not sure whether you are looking for polynomial optimization like Introduction to concepts and advances in polynomial optimization by Martin Mevissen, or polynomial optimization by Hoang Tuy?
answered 8 hours ago
Marco LübbeckeMarco Lübbecke
1,6712 silver badges22 bronze badges
$endgroup$
1
$begingroup$
Thanks for the references Marco. If you have time, it would be great if you could also add some of the main points in the papers for easier future reference.
$endgroup$
– TheSimpliFire
8 hours ago
add a comment |
$begingroup$
+1 for @Marco Lübbecke
But in addition, this is also known as "Polynomial Programming". Also look at algebraic geometry and semialgebraic sets http://www.mit.edu/~parrilo/pubs/files/SDPrelaxations.pdf and https://en.wikipedia.org/wiki/Sum-of-squares_optimization and https://web.stanford.edu/class/ee364b/lectures/sos_slides.pdf. This leads to such cool things as Sum of Squares Programming (optimization), for which Semidefinite Programming relaxation omes into play.
answered 8 hours ago
Mark L. StoneMark L. Stone
2,3495 silver badges23 bronze badges
$endgroup$
add a comment |
$begingroup$
In addition to the excellent answers that are already posted, I want to add that for the pragmatic optimizer, quadratic may already be sufficient.
For example, the cubic constraint $x^3 le 1$ may be replaced by $xy le 1$ and $y=x^2$, which are both quadratic constraints.
This paper is full of examples of non-linear models that can be reformulated as second order cone programs, which are convex quadratic problems. Maximizing a product of non-negative affine functions is one of the examples.
answered 7 hours ago
Kevin DalmeijerKevin Dalmeijer
86315 bronze badges
$endgroup$
$begingroup$
It might be worth noting that $x^3 leq 1$ can be replaced with $x leq 1$, and it might also be worth substituting "SOCP" for "quadratic", since non-convex quadratic problems can be tough to model (they are NP-hard, since the constraint $x_i^2=x_i$ models a binary variable). Assuming that you meant to say SOCP, I agree with your point, since $l_p$ norms are SOCP representable, as proven here.
$endgroup$
– Ryan Cory-Wright
37 mins ago
add a comment |
Your Answer
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
I am not sure whether you are looking for polynomial optimization like Introduction to concepts and advances in polynomial optimization by Martin Mevissen, or polynomial optimization by Hoang Tuy?
answered 8 hours ago
Marco LübbeckeMarco Lübbecke
1,6712 silver badges22 bronze badges
$endgroup$
1
$begingroup$
Thanks for the references Marco. If you have time, it would be great if you could also add some of the main points in the papers for easier future reference.
$endgroup$
– TheSimpliFire
8 hours ago
add a comment |
$begingroup$
I am not sure whether you are looking for polynomial optimization like Introduction to concepts and advances in polynomial optimization by Martin Mevissen, or polynomial optimization by Hoang Tuy?
answered 8 hours ago
Marco LübbeckeMarco Lübbecke
1,6712 silver badges22 bronze badges
$endgroup$
1
$begingroup$
Thanks for the references Marco. If you have time, it would be great if you could also add some of the main points in the papers for easier future reference.
$endgroup$
– TheSimpliFire
8 hours ago
add a comment |
$begingroup$
I am not sure whether you are looking for polynomial optimization like Introduction to concepts and advances in polynomial optimization by Martin Mevissen, or polynomial optimization by Hoang Tuy?
answered 8 hours ago
Marco LübbeckeMarco Lübbecke
1,6712 silver badges22 bronze badges
$endgroup$
I am not sure whether you are looking for polynomial optimization like Introduction to concepts and advances in polynomial optimization by Martin Mevissen, or polynomial optimization by Hoang Tuy?
answered 8 hours ago
Marco LübbeckeMarco Lübbecke
1,6712 silver badges22 bronze badges
edited 7 hours ago
answered 8 hours ago
Marco LübbeckeMarco Lübbecke
1,6712 silver badges22 bronze badges
answered 8 hours ago
Marco LübbeckeMarco Lübbecke
1,6712 silver badges22 bronze badges
answered 8 hours ago
Marco LübbeckeMarco Lübbecke
1,6712 silver badges22 bronze badges
1,6712 silver badges22 bronze badges
1
$begingroup$
Thanks for the references Marco. If you have time, it would be great if you could also add some of the main points in the papers for easier future reference.
$endgroup$
– TheSimpliFire
8 hours ago
add a comment |
1
$begingroup$
Thanks for the references Marco. If you have time, it would be great if you could also add some of the main points in the papers for easier future reference.
$endgroup$
– TheSimpliFire
8 hours ago
1
1
$begingroup$
Thanks for the references Marco. If you have time, it would be great if you could also add some of the main points in the papers for easier future reference.
$endgroup$
– TheSimpliFire
8 hours ago
$begingroup$
Thanks for the references Marco. If you have time, it would be great if you could also add some of the main points in the papers for easier future reference.
$endgroup$
– TheSimpliFire
8 hours ago
add a comment |
$begingroup$
+1 for @Marco Lübbecke
But in addition, this is also known as "Polynomial Programming". Also look at algebraic geometry and semialgebraic sets http://www.mit.edu/~parrilo/pubs/files/SDPrelaxations.pdf and https://en.wikipedia.org/wiki/Sum-of-squares_optimization and https://web.stanford.edu/class/ee364b/lectures/sos_slides.pdf. This leads to such cool things as Sum of Squares Programming (optimization), for which Semidefinite Programming relaxation omes into play.
answered 8 hours ago
Mark L. StoneMark L. Stone
2,3495 silver badges23 bronze badges
$endgroup$
add a comment |
$begingroup$
+1 for @Marco Lübbecke
But in addition, this is also known as "Polynomial Programming". Also look at algebraic geometry and semialgebraic sets http://www.mit.edu/~parrilo/pubs/files/SDPrelaxations.pdf and https://en.wikipedia.org/wiki/Sum-of-squares_optimization and https://web.stanford.edu/class/ee364b/lectures/sos_slides.pdf. This leads to such cool things as Sum of Squares Programming (optimization), for which Semidefinite Programming relaxation omes into play.
answered 8 hours ago
Mark L. StoneMark L. Stone
2,3495 silver badges23 bronze badges
$endgroup$
add a comment |
$begingroup$
+1 for @Marco Lübbecke
But in addition, this is also known as "Polynomial Programming". Also look at algebraic geometry and semialgebraic sets http://www.mit.edu/~parrilo/pubs/files/SDPrelaxations.pdf and https://en.wikipedia.org/wiki/Sum-of-squares_optimization and https://web.stanford.edu/class/ee364b/lectures/sos_slides.pdf. This leads to such cool things as Sum of Squares Programming (optimization), for which Semidefinite Programming relaxation omes into play.
answered 8 hours ago
Mark L. StoneMark L. Stone
2,3495 silver badges23 bronze badges
$endgroup$
+1 for @Marco Lübbecke
But in addition, this is also known as "Polynomial Programming". Also look at algebraic geometry and semialgebraic sets http://www.mit.edu/~parrilo/pubs/files/SDPrelaxations.pdf and https://en.wikipedia.org/wiki/Sum-of-squares_optimization and https://web.stanford.edu/class/ee364b/lectures/sos_slides.pdf. This leads to such cool things as Sum of Squares Programming (optimization), for which Semidefinite Programming relaxation omes into play.
answered 8 hours ago
Mark L. StoneMark L. Stone
2,3495 silver badges23 bronze badges
answered 8 hours ago
Mark L. StoneMark L. Stone
2,3495 silver badges23 bronze badges
answered 8 hours ago
Mark L. StoneMark L. Stone
2,3495 silver badges23 bronze badges
answered 8 hours ago
Mark L. StoneMark L. Stone
2,3495 silver badges23 bronze badges
2,3495 silver badges23 bronze badges
add a comment |
add a comment |
$begingroup$
In addition to the excellent answers that are already posted, I want to add that for the pragmatic optimizer, quadratic may already be sufficient.
For example, the cubic constraint $x^3 le 1$ may be replaced by $xy le 1$ and $y=x^2$, which are both quadratic constraints.
This paper is full of examples of non-linear models that can be reformulated as second order cone programs, which are convex quadratic problems. Maximizing a product of non-negative affine functions is one of the examples.
answered 7 hours ago
Kevin DalmeijerKevin Dalmeijer
86315 bronze badges
$endgroup$
$begingroup$
It might be worth noting that $x^3 leq 1$ can be replaced with $x leq 1$, and it might also be worth substituting "SOCP" for "quadratic", since non-convex quadratic problems can be tough to model (they are NP-hard, since the constraint $x_i^2=x_i$ models a binary variable). Assuming that you meant to say SOCP, I agree with your point, since $l_p$ norms are SOCP representable, as proven here.
$endgroup$
– Ryan Cory-Wright
37 mins ago
add a comment |
$begingroup$
In addition to the excellent answers that are already posted, I want to add that for the pragmatic optimizer, quadratic may already be sufficient.
For example, the cubic constraint $x^3 le 1$ may be replaced by $xy le 1$ and $y=x^2$, which are both quadratic constraints.
This paper is full of examples of non-linear models that can be reformulated as second order cone programs, which are convex quadratic problems. Maximizing a product of non-negative affine functions is one of the examples.
answered 7 hours ago
Kevin DalmeijerKevin Dalmeijer
86315 bronze badges
$endgroup$
$begingroup$
It might be worth noting that $x^3 leq 1$ can be replaced with $x leq 1$, and it might also be worth substituting "SOCP" for "quadratic", since non-convex quadratic problems can be tough to model (they are NP-hard, since the constraint $x_i^2=x_i$ models a binary variable). Assuming that you meant to say SOCP, I agree with your point, since $l_p$ norms are SOCP representable, as proven here.
$endgroup$
– Ryan Cory-Wright
37 mins ago
add a comment |
$begingroup$
In addition to the excellent answers that are already posted, I want to add that for the pragmatic optimizer, quadratic may already be sufficient.
For example, the cubic constraint $x^3 le 1$ may be replaced by $xy le 1$ and $y=x^2$, which are both quadratic constraints.
This paper is full of examples of non-linear models that can be reformulated as second order cone programs, which are convex quadratic problems. Maximizing a product of non-negative affine functions is one of the examples.
answered 7 hours ago
Kevin DalmeijerKevin Dalmeijer
86315 bronze badges
$endgroup$
In addition to the excellent answers that are already posted, I want to add that for the pragmatic optimizer, quadratic may already be sufficient.
For example, the cubic constraint $x^3 le 1$ may be replaced by $xy le 1$ and $y=x^2$, which are both quadratic constraints.
This paper is full of examples of non-linear models that can be reformulated as second order cone programs, which are convex quadratic problems. Maximizing a product of non-negative affine functions is one of the examples.
answered 7 hours ago
Kevin DalmeijerKevin Dalmeijer
86315 bronze badges
answered 7 hours ago
Kevin DalmeijerKevin Dalmeijer
86315 bronze badges
answered 7 hours ago
Kevin DalmeijerKevin Dalmeijer
86315 bronze badges
answered 7 hours ago
Kevin DalmeijerKevin Dalmeijer
86315 bronze badges
86315 bronze badges
$begingroup$
It might be worth noting that $x^3 leq 1$ can be replaced with $x leq 1$, and it might also be worth substituting "SOCP" for "quadratic", since non-convex quadratic problems can be tough to model (they are NP-hard, since the constraint $x_i^2=x_i$ models a binary variable). Assuming that you meant to say SOCP, I agree with your point, since $l_p$ norms are SOCP representable, as proven here.
$endgroup$
– Ryan Cory-Wright
37 mins ago
add a comment |
$begingroup$
It might be worth noting that $x^3 leq 1$ can be replaced with $x leq 1$, and it might also be worth substituting "SOCP" for "quadratic", since non-convex quadratic problems can be tough to model (they are NP-hard, since the constraint $x_i^2=x_i$ models a binary variable). Assuming that you meant to say SOCP, I agree with your point, since $l_p$ norms are SOCP representable, as proven here.
$endgroup$
– Ryan Cory-Wright
37 mins ago
$begingroup$
It might be worth noting that $x^3 leq 1$ can be replaced with $x leq 1$, and it might also be worth substituting "SOCP" for "quadratic", since non-convex quadratic problems can be tough to model (they are NP-hard, since the constraint $x_i^2=x_i$ models a binary variable). Assuming that you meant to say SOCP, I agree with your point, since $l_p$ norms are SOCP representable, as proven here.
$endgroup$
– Ryan Cory-Wright
37 mins ago
$begingroup$
It might be worth noting that $x^3 leq 1$ can be replaced with $x leq 1$, and it might also be worth substituting "SOCP" for "quadratic", since non-convex quadratic problems can be tough to model (they are NP-hard, since the constraint $x_i^2=x_i$ models a binary variable). Assuming that you meant to say SOCP, I agree with your point, since $l_p$ norms are SOCP representable, as proven here.
$endgroup$
– Ryan Cory-Wright
37 mins ago
add a comment |
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