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How to determine the convexity of my problem and categorize it?
Solving ATSP problem for large-scale problemWhat is the connection of Operations Research and Reinforcement Learning?How to determine if a given problem seems to be a good fit to be solved using combinatorial Benders decompositionHow to handle real-world (soft) constraints in an optimization problem?Bound on the number of constraints to be generated (lazy constraints)How can I linearize or convexify this binary quadratic optimization problem?How to formulate this scheduling problem efficiently?What is a “hard problem” in the context of Mixed-integer programming?
$begingroup$
My problem is:
beginalignminlimits_x_ijqquad&sum_iin Nsum_jin Mfracx_ijC_j-sumlimits_iin N x_ija_i\texts.t.qquad&0<C_j-sum_iin N x_ija_i\qquad&sum_jin M x_ij=1\qquad&x_ijin[0,1]\quad&sum_jin Mfracx_ijC_j-sumlimits_iin N x_ija_i le d_i,forall iin Nendalign
It is worth mentioning that the third constraint indicates the range and $x_ij$ isn't binary. As I calculated the Hessian of my objective function, I understood that the sign of matrix elements is dependent on $x_ij$ and $C_j-sumlimits_iin N x_ija_i$ which are always positive or zero.
- Due to this, can I conclude that my problem is convex?
- If the answer is yes, what class of convex problems does it belong to? (conic, geometric, etc.) and if no what is the type of problem?
optimization convexity
asked 8 hours ago
Benyamin TBenyamin T
285 bronze badges
New contributor
Benyamin T is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
My problem is:
beginalignminlimits_x_ijqquad&sum_iin Nsum_jin Mfracx_ijC_j-sumlimits_iin N x_ija_i\texts.t.qquad&0<C_j-sum_iin N x_ija_i\qquad&sum_jin M x_ij=1\qquad&x_ijin[0,1]\quad&sum_jin Mfracx_ijC_j-sumlimits_iin N x_ija_i le d_i,forall iin Nendalign
It is worth mentioning that the third constraint indicates the range and $x_ij$ isn't binary. As I calculated the Hessian of my objective function, I understood that the sign of matrix elements is dependent on $x_ij$ and $C_j-sumlimits_iin N x_ija_i$ which are always positive or zero.
- Due to this, can I conclude that my problem is convex?
- If the answer is yes, what class of convex problems does it belong to? (conic, geometric, etc.) and if no what is the type of problem?
optimization convexity
asked 8 hours ago
Benyamin TBenyamin T
285 bronze badges
New contributor
Benyamin T is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
2
$begingroup$
You are using $i$ in three different and conflicting ways: as an outer summation index, an inner summation index, and a constraint index.
$endgroup$
– Rob Pratt
7 hours ago
$begingroup$
Your last constraint sums over $i$ on the left but has a RHS indexed by $i$. This cannot be parsed.
$endgroup$
– prubin
1 hour ago
$begingroup$
@prubin Thanks, I corrected it.
$endgroup$
– Benyamin T
52 mins ago
add a comment |
$begingroup$
My problem is:
beginalignminlimits_x_ijqquad&sum_iin Nsum_jin Mfracx_ijC_j-sumlimits_iin N x_ija_i\texts.t.qquad&0<C_j-sum_iin N x_ija_i\qquad&sum_jin M x_ij=1\qquad&x_ijin[0,1]\quad&sum_jin Mfracx_ijC_j-sumlimits_iin N x_ija_i le d_i,forall iin Nendalign
It is worth mentioning that the third constraint indicates the range and $x_ij$ isn't binary. As I calculated the Hessian of my objective function, I understood that the sign of matrix elements is dependent on $x_ij$ and $C_j-sumlimits_iin N x_ija_i$ which are always positive or zero.
- Due to this, can I conclude that my problem is convex?
- If the answer is yes, what class of convex problems does it belong to? (conic, geometric, etc.) and if no what is the type of problem?
optimization convexity
asked 8 hours ago
Benyamin TBenyamin T
285 bronze badges
New contributor
Benyamin T is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
My problem is:
beginalignminlimits_x_ijqquad&sum_iin Nsum_jin Mfracx_ijC_j-sumlimits_iin N x_ija_i\texts.t.qquad&0<C_j-sum_iin N x_ija_i\qquad&sum_jin M x_ij=1\qquad&x_ijin[0,1]\quad&sum_jin Mfracx_ijC_j-sumlimits_iin N x_ija_i le d_i,forall iin Nendalign
It is worth mentioning that the third constraint indicates the range and $x_ij$ isn't binary. As I calculated the Hessian of my objective function, I understood that the sign of matrix elements is dependent on $x_ij$ and $C_j-sumlimits_iin N x_ija_i$ which are always positive or zero.
- Due to this, can I conclude that my problem is convex?
- If the answer is yes, what class of convex problems does it belong to? (conic, geometric, etc.) and if no what is the type of problem?
optimization convexity
optimization convexity
asked 8 hours ago
Benyamin TBenyamin T
285 bronze badges
New contributor
Benyamin T is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 8 hours ago
Benyamin TBenyamin T
285 bronze badges
New contributor
Benyamin T is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 53 mins ago
Benyamin T
asked 8 hours ago
Benyamin TBenyamin T
285 bronze badges
New contributor
Benyamin T is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 8 hours ago
Benyamin TBenyamin T
285 bronze badges
asked 8 hours ago
Benyamin TBenyamin T
285 bronze badges
285 bronze badges
New contributor
Benyamin T is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Benyamin T is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
2
$begingroup$
You are using $i$ in three different and conflicting ways: as an outer summation index, an inner summation index, and a constraint index.
$endgroup$
– Rob Pratt
7 hours ago
$begingroup$
Your last constraint sums over $i$ on the left but has a RHS indexed by $i$. This cannot be parsed.
$endgroup$
– prubin
1 hour ago
$begingroup$
@prubin Thanks, I corrected it.
$endgroup$
– Benyamin T
52 mins ago
add a comment |
2
$begingroup$
You are using $i$ in three different and conflicting ways: as an outer summation index, an inner summation index, and a constraint index.
$endgroup$
– Rob Pratt
7 hours ago
$begingroup$
Your last constraint sums over $i$ on the left but has a RHS indexed by $i$. This cannot be parsed.
$endgroup$
– prubin
1 hour ago
$begingroup$
@prubin Thanks, I corrected it.
$endgroup$
– Benyamin T
52 mins ago
2
2
$begingroup$
You are using $i$ in three different and conflicting ways: as an outer summation index, an inner summation index, and a constraint index.
$endgroup$
– Rob Pratt
7 hours ago
$begingroup$
You are using $i$ in three different and conflicting ways: as an outer summation index, an inner summation index, and a constraint index.
$endgroup$
– Rob Pratt
7 hours ago
$begingroup$
Your last constraint sums over $i$ on the left but has a RHS indexed by $i$. This cannot be parsed.
$endgroup$
– prubin
1 hour ago
$begingroup$
Your last constraint sums over $i$ on the left but has a RHS indexed by $i$. This cannot be parsed.
$endgroup$
– prubin
1 hour ago
$begingroup$
@prubin Thanks, I corrected it.
$endgroup$
– Benyamin T
52 mins ago
$begingroup$
@prubin Thanks, I corrected it.
$endgroup$
– Benyamin T
52 mins ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
This is a Linear-Fractional Programming problem.
It can be transformed to a Linear Programming problem as shown in section 4.3.2 "Linear-fractional programming" of "Convex Optimization" by Boyd and Vandenberghe
answered 8 hours ago
Mark L. StoneMark L. Stone
3,4127 silver badges27 bronze badges
$endgroup$
1
$begingroup$
Just to be clear, the stated problem is actually a generalization of the linear-fractional problem in which there is a linear sun of fractions (sum wiith respect to j). So the transformation in the linked section will have to be applied using new variables, and applied separately, for each fraction (each value of j) in the sum. Everything still works out.
$endgroup$
– Mark L. Stone
7 hours ago
$begingroup$
Thanks for your helpful answer.
$endgroup$
– Benyamin T
51 mins ago
add a comment |
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1 Answer
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$begingroup$
This is a Linear-Fractional Programming problem.
It can be transformed to a Linear Programming problem as shown in section 4.3.2 "Linear-fractional programming" of "Convex Optimization" by Boyd and Vandenberghe
answered 8 hours ago
Mark L. StoneMark L. Stone
3,4127 silver badges27 bronze badges
$endgroup$
1
$begingroup$
Just to be clear, the stated problem is actually a generalization of the linear-fractional problem in which there is a linear sun of fractions (sum wiith respect to j). So the transformation in the linked section will have to be applied using new variables, and applied separately, for each fraction (each value of j) in the sum. Everything still works out.
$endgroup$
– Mark L. Stone
7 hours ago
$begingroup$
Thanks for your helpful answer.
$endgroup$
– Benyamin T
51 mins ago
add a comment |
$begingroup$
This is a Linear-Fractional Programming problem.
It can be transformed to a Linear Programming problem as shown in section 4.3.2 "Linear-fractional programming" of "Convex Optimization" by Boyd and Vandenberghe
answered 8 hours ago
Mark L. StoneMark L. Stone
3,4127 silver badges27 bronze badges
$endgroup$
1
$begingroup$
Just to be clear, the stated problem is actually a generalization of the linear-fractional problem in which there is a linear sun of fractions (sum wiith respect to j). So the transformation in the linked section will have to be applied using new variables, and applied separately, for each fraction (each value of j) in the sum. Everything still works out.
$endgroup$
– Mark L. Stone
7 hours ago
$begingroup$
Thanks for your helpful answer.
$endgroup$
– Benyamin T
51 mins ago
add a comment |
$begingroup$
This is a Linear-Fractional Programming problem.
It can be transformed to a Linear Programming problem as shown in section 4.3.2 "Linear-fractional programming" of "Convex Optimization" by Boyd and Vandenberghe
answered 8 hours ago
Mark L. StoneMark L. Stone
3,4127 silver badges27 bronze badges
$endgroup$
This is a Linear-Fractional Programming problem.
It can be transformed to a Linear Programming problem as shown in section 4.3.2 "Linear-fractional programming" of "Convex Optimization" by Boyd and Vandenberghe
answered 8 hours ago
Mark L. StoneMark L. Stone
3,4127 silver badges27 bronze badges
answered 8 hours ago
Mark L. StoneMark L. Stone
3,4127 silver badges27 bronze badges
answered 8 hours ago
Mark L. StoneMark L. Stone
3,4127 silver badges27 bronze badges
answered 8 hours ago
Mark L. StoneMark L. Stone
3,4127 silver badges27 bronze badges
3,4127 silver badges27 bronze badges
1
$begingroup$
Just to be clear, the stated problem is actually a generalization of the linear-fractional problem in which there is a linear sun of fractions (sum wiith respect to j). So the transformation in the linked section will have to be applied using new variables, and applied separately, for each fraction (each value of j) in the sum. Everything still works out.
$endgroup$
– Mark L. Stone
7 hours ago
$begingroup$
Thanks for your helpful answer.
$endgroup$
– Benyamin T
51 mins ago
add a comment |
1
$begingroup$
Just to be clear, the stated problem is actually a generalization of the linear-fractional problem in which there is a linear sun of fractions (sum wiith respect to j). So the transformation in the linked section will have to be applied using new variables, and applied separately, for each fraction (each value of j) in the sum. Everything still works out.
$endgroup$
– Mark L. Stone
7 hours ago
$begingroup$
Thanks for your helpful answer.
$endgroup$
– Benyamin T
51 mins ago
1
1
$begingroup$
Just to be clear, the stated problem is actually a generalization of the linear-fractional problem in which there is a linear sun of fractions (sum wiith respect to j). So the transformation in the linked section will have to be applied using new variables, and applied separately, for each fraction (each value of j) in the sum. Everything still works out.
$endgroup$
– Mark L. Stone
7 hours ago
$begingroup$
Just to be clear, the stated problem is actually a generalization of the linear-fractional problem in which there is a linear sun of fractions (sum wiith respect to j). So the transformation in the linked section will have to be applied using new variables, and applied separately, for each fraction (each value of j) in the sum. Everything still works out.
$endgroup$
– Mark L. Stone
7 hours ago
$begingroup$
Thanks for your helpful answer.
$endgroup$
– Benyamin T
51 mins ago
$begingroup$
Thanks for your helpful answer.
$endgroup$
– Benyamin T
51 mins ago
add a comment |
Benyamin T is a new contributor. Be nice, and check out our Code of Conduct.
Benyamin T is a new contributor. Be nice, and check out our Code of Conduct.
Benyamin T is a new contributor. Be nice, and check out our Code of Conduct.
Benyamin T is a new contributor. Be nice, and check out our Code of Conduct.
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2
$begingroup$
You are using $i$ in three different and conflicting ways: as an outer summation index, an inner summation index, and a constraint index.
$endgroup$
– Rob Pratt
7 hours ago
$begingroup$
Your last constraint sums over $i$ on the left but has a RHS indexed by $i$. This cannot be parsed.
$endgroup$
– prubin
1 hour ago
$begingroup$
@prubin Thanks, I corrected it.
$endgroup$
– Benyamin T
52 mins ago