Iterative algorithm for a simple linear optimization problemAn Optimization problem related with $(-1)^N-1sum_i=1^Mfracln x_ix_i^Nprod_jneq ifracx_ix_i-x_j$Optimization problem with a minimization sub-problem as a constraintProving boundedness of a function .How is the pivot chosen for the symbolic weights for the Cassowary algorithm?What happens if we remove the non-negativity constraints in a linear programming problem?An optimization problem with strict inequalities for which an optimal solution existsSolving a convex resource allocation problemhow can update parameter in each iteration?Dual of a linear program.Binary Polymatroid Optimization Problem

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Iterative algorithm for a simple linear optimization problem


An Optimization problem related with $(-1)^N-1sum_i=1^Mfracln x_ix_i^Nprod_jneq ifracx_ix_i-x_j$Optimization problem with a minimization sub-problem as a constraintProving boundedness of a function .How is the pivot chosen for the symbolic weights for the Cassowary algorithm?What happens if we remove the non-negativity constraints in a linear programming problem?An optimization problem with strict inequalities for which an optimal solution existsSolving a convex resource allocation problemhow can update parameter in each iteration?Dual of a linear program.Binary Polymatroid Optimization Problem













1












$begingroup$


Let $c_1,dots,c_n$ be $n$ positive numbers and so be $a_1,dots, a_n$. For some $r$ such that $1leq rleq n$, consider the optimization problem
beginalign
max_x_iinmathbbR&~~sum_i=1^rc_ix_i - sum_i=r+1^nc_ix_i \
& sum_i=1^ra_ix_i - sum_i=r+1^na_ix_i leq b \
& 0leq x_i leq 1~,~forall i
endalign

I understand that I can use a linear solver for this, however I am interested in finding a separate algorithm for this. I am using the following approach to solve it. First I initialize all $x_i$ as
beginalign
x_i , = , begincases1 ~~textif~iin1,dots,r\ 0 ~~textif~iinr+1,dots,nendcases
endalign

Note that this is the objective value at the optimum for the unconstrained problem which will be always larger (or equal) than the objective value at optimum of the constrained one. At this point, if this point is feasible, then we are done. Else, note that $c_i_r+1^n$ all have a decreasing effect on both objective and LHS of constraint (making it more likely to satisfy the constraint). Thus, I pick the smallest among these $c_i$ and increases its $x_i$ in $[0,1]$ to see if the constraint becomes feasible while decreasing the objective. If not, set $x_i=1$ and pick the second smallest in $x_i_i=r+1^n$ and continue in this fashion till I exhaust all in that set. Then I move on to the set $c_i_1^r$ and do similar operations there. My question is whether this will find the true solution?










share|cite|improve this question











$endgroup$











  • $begingroup$
    I believe that constructing a one-dimensional Lagrangian dual problem is your way to construct a specialized algorithm. Construct a Lagrangian with a multiplier $lambda > 0$ for the first inequality constraint, and minimize it subject to $x_i in [0, 1]$. You can maximize it any one-dimensional method you like, such as Golden Section search, and then extract the primal optimal by substiuting $lambda^*$ into the Lagrangian minimization problem.
    $endgroup$
    – Alex Shtof
    Mar 17 at 8:09










  • $begingroup$
    Please clear this doubt. Is skipping the multipliers corresponding to $x_iin [0,1]$ is ok? because they are also constraints, right?
    $endgroup$
    – dineshdileep
    Mar 17 at 12:04











  • $begingroup$
    Yes. Duality works for convex problems of the form $min~f(x)~texts.t.~ g_i(x) leq 0~ x in C$. The dual function is $q(lambda) = min_x in C f(x) + sum_i lambda_i g_i(x) $. It is our choice which constraints we treat as $g_i$ and which constraints we treat as $C$. I chose to treat $x_i in [0, 1]$ as the set $C$, and left with only one constraint which has a multiplier.
    $endgroup$
    – Alex Shtof
    Mar 17 at 12:13











  • $begingroup$
    Got, it. Thanks!.
    $endgroup$
    – dineshdileep
    Mar 17 at 12:34










  • $begingroup$
    @AlexShtof how are KKT conditions satisfied here?
    $endgroup$
    – dineshdileep
    Mar 17 at 21:57















1












$begingroup$


Let $c_1,dots,c_n$ be $n$ positive numbers and so be $a_1,dots, a_n$. For some $r$ such that $1leq rleq n$, consider the optimization problem
beginalign
max_x_iinmathbbR&~~sum_i=1^rc_ix_i - sum_i=r+1^nc_ix_i \
& sum_i=1^ra_ix_i - sum_i=r+1^na_ix_i leq b \
& 0leq x_i leq 1~,~forall i
endalign

I understand that I can use a linear solver for this, however I am interested in finding a separate algorithm for this. I am using the following approach to solve it. First I initialize all $x_i$ as
beginalign
x_i , = , begincases1 ~~textif~iin1,dots,r\ 0 ~~textif~iinr+1,dots,nendcases
endalign

Note that this is the objective value at the optimum for the unconstrained problem which will be always larger (or equal) than the objective value at optimum of the constrained one. At this point, if this point is feasible, then we are done. Else, note that $c_i_r+1^n$ all have a decreasing effect on both objective and LHS of constraint (making it more likely to satisfy the constraint). Thus, I pick the smallest among these $c_i$ and increases its $x_i$ in $[0,1]$ to see if the constraint becomes feasible while decreasing the objective. If not, set $x_i=1$ and pick the second smallest in $x_i_i=r+1^n$ and continue in this fashion till I exhaust all in that set. Then I move on to the set $c_i_1^r$ and do similar operations there. My question is whether this will find the true solution?










share|cite|improve this question











$endgroup$











  • $begingroup$
    I believe that constructing a one-dimensional Lagrangian dual problem is your way to construct a specialized algorithm. Construct a Lagrangian with a multiplier $lambda > 0$ for the first inequality constraint, and minimize it subject to $x_i in [0, 1]$. You can maximize it any one-dimensional method you like, such as Golden Section search, and then extract the primal optimal by substiuting $lambda^*$ into the Lagrangian minimization problem.
    $endgroup$
    – Alex Shtof
    Mar 17 at 8:09










  • $begingroup$
    Please clear this doubt. Is skipping the multipliers corresponding to $x_iin [0,1]$ is ok? because they are also constraints, right?
    $endgroup$
    – dineshdileep
    Mar 17 at 12:04











  • $begingroup$
    Yes. Duality works for convex problems of the form $min~f(x)~texts.t.~ g_i(x) leq 0~ x in C$. The dual function is $q(lambda) = min_x in C f(x) + sum_i lambda_i g_i(x) $. It is our choice which constraints we treat as $g_i$ and which constraints we treat as $C$. I chose to treat $x_i in [0, 1]$ as the set $C$, and left with only one constraint which has a multiplier.
    $endgroup$
    – Alex Shtof
    Mar 17 at 12:13











  • $begingroup$
    Got, it. Thanks!.
    $endgroup$
    – dineshdileep
    Mar 17 at 12:34










  • $begingroup$
    @AlexShtof how are KKT conditions satisfied here?
    $endgroup$
    – dineshdileep
    Mar 17 at 21:57













1












1








1





$begingroup$


Let $c_1,dots,c_n$ be $n$ positive numbers and so be $a_1,dots, a_n$. For some $r$ such that $1leq rleq n$, consider the optimization problem
beginalign
max_x_iinmathbbR&~~sum_i=1^rc_ix_i - sum_i=r+1^nc_ix_i \
& sum_i=1^ra_ix_i - sum_i=r+1^na_ix_i leq b \
& 0leq x_i leq 1~,~forall i
endalign

I understand that I can use a linear solver for this, however I am interested in finding a separate algorithm for this. I am using the following approach to solve it. First I initialize all $x_i$ as
beginalign
x_i , = , begincases1 ~~textif~iin1,dots,r\ 0 ~~textif~iinr+1,dots,nendcases
endalign

Note that this is the objective value at the optimum for the unconstrained problem which will be always larger (or equal) than the objective value at optimum of the constrained one. At this point, if this point is feasible, then we are done. Else, note that $c_i_r+1^n$ all have a decreasing effect on both objective and LHS of constraint (making it more likely to satisfy the constraint). Thus, I pick the smallest among these $c_i$ and increases its $x_i$ in $[0,1]$ to see if the constraint becomes feasible while decreasing the objective. If not, set $x_i=1$ and pick the second smallest in $x_i_i=r+1^n$ and continue in this fashion till I exhaust all in that set. Then I move on to the set $c_i_1^r$ and do similar operations there. My question is whether this will find the true solution?










share|cite|improve this question











$endgroup$




Let $c_1,dots,c_n$ be $n$ positive numbers and so be $a_1,dots, a_n$. For some $r$ such that $1leq rleq n$, consider the optimization problem
beginalign
max_x_iinmathbbR&~~sum_i=1^rc_ix_i - sum_i=r+1^nc_ix_i \
& sum_i=1^ra_ix_i - sum_i=r+1^na_ix_i leq b \
& 0leq x_i leq 1~,~forall i
endalign

I understand that I can use a linear solver for this, however I am interested in finding a separate algorithm for this. I am using the following approach to solve it. First I initialize all $x_i$ as
beginalign
x_i , = , begincases1 ~~textif~iin1,dots,r\ 0 ~~textif~iinr+1,dots,nendcases
endalign

Note that this is the objective value at the optimum for the unconstrained problem which will be always larger (or equal) than the objective value at optimum of the constrained one. At this point, if this point is feasible, then we are done. Else, note that $c_i_r+1^n$ all have a decreasing effect on both objective and LHS of constraint (making it more likely to satisfy the constraint). Thus, I pick the smallest among these $c_i$ and increases its $x_i$ in $[0,1]$ to see if the constraint becomes feasible while decreasing the objective. If not, set $x_i=1$ and pick the second smallest in $x_i_i=r+1^n$ and continue in this fashion till I exhaust all in that set. Then I move on to the set $c_i_1^r$ and do similar operations there. My question is whether this will find the true solution?







real-analysis linear-algebra optimization linear-programming numerical-optimization






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Mar 16 at 12:01









Rodrigo de Azevedo

13.2k41960




13.2k41960










asked Mar 16 at 11:58









dineshdileepdineshdileep

6,00111935




6,00111935











  • $begingroup$
    I believe that constructing a one-dimensional Lagrangian dual problem is your way to construct a specialized algorithm. Construct a Lagrangian with a multiplier $lambda > 0$ for the first inequality constraint, and minimize it subject to $x_i in [0, 1]$. You can maximize it any one-dimensional method you like, such as Golden Section search, and then extract the primal optimal by substiuting $lambda^*$ into the Lagrangian minimization problem.
    $endgroup$
    – Alex Shtof
    Mar 17 at 8:09










  • $begingroup$
    Please clear this doubt. Is skipping the multipliers corresponding to $x_iin [0,1]$ is ok? because they are also constraints, right?
    $endgroup$
    – dineshdileep
    Mar 17 at 12:04











  • $begingroup$
    Yes. Duality works for convex problems of the form $min~f(x)~texts.t.~ g_i(x) leq 0~ x in C$. The dual function is $q(lambda) = min_x in C f(x) + sum_i lambda_i g_i(x) $. It is our choice which constraints we treat as $g_i$ and which constraints we treat as $C$. I chose to treat $x_i in [0, 1]$ as the set $C$, and left with only one constraint which has a multiplier.
    $endgroup$
    – Alex Shtof
    Mar 17 at 12:13











  • $begingroup$
    Got, it. Thanks!.
    $endgroup$
    – dineshdileep
    Mar 17 at 12:34










  • $begingroup$
    @AlexShtof how are KKT conditions satisfied here?
    $endgroup$
    – dineshdileep
    Mar 17 at 21:57
















  • $begingroup$
    I believe that constructing a one-dimensional Lagrangian dual problem is your way to construct a specialized algorithm. Construct a Lagrangian with a multiplier $lambda > 0$ for the first inequality constraint, and minimize it subject to $x_i in [0, 1]$. You can maximize it any one-dimensional method you like, such as Golden Section search, and then extract the primal optimal by substiuting $lambda^*$ into the Lagrangian minimization problem.
    $endgroup$
    – Alex Shtof
    Mar 17 at 8:09










  • $begingroup$
    Please clear this doubt. Is skipping the multipliers corresponding to $x_iin [0,1]$ is ok? because they are also constraints, right?
    $endgroup$
    – dineshdileep
    Mar 17 at 12:04











  • $begingroup$
    Yes. Duality works for convex problems of the form $min~f(x)~texts.t.~ g_i(x) leq 0~ x in C$. The dual function is $q(lambda) = min_x in C f(x) + sum_i lambda_i g_i(x) $. It is our choice which constraints we treat as $g_i$ and which constraints we treat as $C$. I chose to treat $x_i in [0, 1]$ as the set $C$, and left with only one constraint which has a multiplier.
    $endgroup$
    – Alex Shtof
    Mar 17 at 12:13











  • $begingroup$
    Got, it. Thanks!.
    $endgroup$
    – dineshdileep
    Mar 17 at 12:34










  • $begingroup$
    @AlexShtof how are KKT conditions satisfied here?
    $endgroup$
    – dineshdileep
    Mar 17 at 21:57















$begingroup$
I believe that constructing a one-dimensional Lagrangian dual problem is your way to construct a specialized algorithm. Construct a Lagrangian with a multiplier $lambda > 0$ for the first inequality constraint, and minimize it subject to $x_i in [0, 1]$. You can maximize it any one-dimensional method you like, such as Golden Section search, and then extract the primal optimal by substiuting $lambda^*$ into the Lagrangian minimization problem.
$endgroup$
– Alex Shtof
Mar 17 at 8:09




$begingroup$
I believe that constructing a one-dimensional Lagrangian dual problem is your way to construct a specialized algorithm. Construct a Lagrangian with a multiplier $lambda > 0$ for the first inequality constraint, and minimize it subject to $x_i in [0, 1]$. You can maximize it any one-dimensional method you like, such as Golden Section search, and then extract the primal optimal by substiuting $lambda^*$ into the Lagrangian minimization problem.
$endgroup$
– Alex Shtof
Mar 17 at 8:09












$begingroup$
Please clear this doubt. Is skipping the multipliers corresponding to $x_iin [0,1]$ is ok? because they are also constraints, right?
$endgroup$
– dineshdileep
Mar 17 at 12:04





$begingroup$
Please clear this doubt. Is skipping the multipliers corresponding to $x_iin [0,1]$ is ok? because they are also constraints, right?
$endgroup$
– dineshdileep
Mar 17 at 12:04













$begingroup$
Yes. Duality works for convex problems of the form $min~f(x)~texts.t.~ g_i(x) leq 0~ x in C$. The dual function is $q(lambda) = min_x in C f(x) + sum_i lambda_i g_i(x) $. It is our choice which constraints we treat as $g_i$ and which constraints we treat as $C$. I chose to treat $x_i in [0, 1]$ as the set $C$, and left with only one constraint which has a multiplier.
$endgroup$
– Alex Shtof
Mar 17 at 12:13





$begingroup$
Yes. Duality works for convex problems of the form $min~f(x)~texts.t.~ g_i(x) leq 0~ x in C$. The dual function is $q(lambda) = min_x in C f(x) + sum_i lambda_i g_i(x) $. It is our choice which constraints we treat as $g_i$ and which constraints we treat as $C$. I chose to treat $x_i in [0, 1]$ as the set $C$, and left with only one constraint which has a multiplier.
$endgroup$
– Alex Shtof
Mar 17 at 12:13













$begingroup$
Got, it. Thanks!.
$endgroup$
– dineshdileep
Mar 17 at 12:34




$begingroup$
Got, it. Thanks!.
$endgroup$
– dineshdileep
Mar 17 at 12:34












$begingroup$
@AlexShtof how are KKT conditions satisfied here?
$endgroup$
– dineshdileep
Mar 17 at 21:57




$begingroup$
@AlexShtof how are KKT conditions satisfied here?
$endgroup$
– dineshdileep
Mar 17 at 21:57










1 Answer
1






active

oldest

votes


















0












$begingroup$

This algorithm is suboptimal. Suppose $a_i=0$ then you change $x_i$ unnecessarily. If you consider the ratio of $c_i$ and $a_i$, you almost mimicked the simplex method.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Can you explain the part on simplex method more?
    $endgroup$
    – dineshdileep
    Mar 16 at 13:54










  • $begingroup$
    @dineshdileep I would need some time to work out the details and I am not 100% certain it boils down to the same result, but the ratio test in the dual problem considers the same ratios.
    $endgroup$
    – LinAlg
    Mar 16 at 14:20










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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









0












$begingroup$

This algorithm is suboptimal. Suppose $a_i=0$ then you change $x_i$ unnecessarily. If you consider the ratio of $c_i$ and $a_i$, you almost mimicked the simplex method.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Can you explain the part on simplex method more?
    $endgroup$
    – dineshdileep
    Mar 16 at 13:54










  • $begingroup$
    @dineshdileep I would need some time to work out the details and I am not 100% certain it boils down to the same result, but the ratio test in the dual problem considers the same ratios.
    $endgroup$
    – LinAlg
    Mar 16 at 14:20















0












$begingroup$

This algorithm is suboptimal. Suppose $a_i=0$ then you change $x_i$ unnecessarily. If you consider the ratio of $c_i$ and $a_i$, you almost mimicked the simplex method.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Can you explain the part on simplex method more?
    $endgroup$
    – dineshdileep
    Mar 16 at 13:54










  • $begingroup$
    @dineshdileep I would need some time to work out the details and I am not 100% certain it boils down to the same result, but the ratio test in the dual problem considers the same ratios.
    $endgroup$
    – LinAlg
    Mar 16 at 14:20













0












0








0





$begingroup$

This algorithm is suboptimal. Suppose $a_i=0$ then you change $x_i$ unnecessarily. If you consider the ratio of $c_i$ and $a_i$, you almost mimicked the simplex method.






share|cite|improve this answer









$endgroup$



This algorithm is suboptimal. Suppose $a_i=0$ then you change $x_i$ unnecessarily. If you consider the ratio of $c_i$ and $a_i$, you almost mimicked the simplex method.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Mar 16 at 13:37









LinAlgLinAlg

10.1k1521




10.1k1521











  • $begingroup$
    Can you explain the part on simplex method more?
    $endgroup$
    – dineshdileep
    Mar 16 at 13:54










  • $begingroup$
    @dineshdileep I would need some time to work out the details and I am not 100% certain it boils down to the same result, but the ratio test in the dual problem considers the same ratios.
    $endgroup$
    – LinAlg
    Mar 16 at 14:20
















  • $begingroup$
    Can you explain the part on simplex method more?
    $endgroup$
    – dineshdileep
    Mar 16 at 13:54










  • $begingroup$
    @dineshdileep I would need some time to work out the details and I am not 100% certain it boils down to the same result, but the ratio test in the dual problem considers the same ratios.
    $endgroup$
    – LinAlg
    Mar 16 at 14:20















$begingroup$
Can you explain the part on simplex method more?
$endgroup$
– dineshdileep
Mar 16 at 13:54




$begingroup$
Can you explain the part on simplex method more?
$endgroup$
– dineshdileep
Mar 16 at 13:54












$begingroup$
@dineshdileep I would need some time to work out the details and I am not 100% certain it boils down to the same result, but the ratio test in the dual problem considers the same ratios.
$endgroup$
– LinAlg
Mar 16 at 14:20




$begingroup$
@dineshdileep I would need some time to work out the details and I am not 100% certain it boils down to the same result, but the ratio test in the dual problem considers the same ratios.
$endgroup$
– LinAlg
Mar 16 at 14:20

















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Kathakali Contents Etymology and nomenclature History Repertoire Songs and musical instruments Traditional plays Styles: Sampradayam Training centers and awards Relationship to other dance forms See also Notes References External links Navigation menueThe Illustrated Encyclopedia of Hinduism: A-MSouth Asian Folklore: An EncyclopediaRoutledge International Encyclopedia of Women: Global Women's Issues and KnowledgeKathakali Dance-drama: Where Gods and Demons Come to PlayKathakali Dance-drama: Where Gods and Demons Come to PlayKathakali Dance-drama: Where Gods and Demons Come to Play10.1353/atj.2005.0004The Illustrated Encyclopedia of Hinduism: A-MEncyclopedia of HinduismKathakali Dance-drama: Where Gods and Demons Come to PlaySonic Liturgy: Ritual and Music in Hindu Tradition"The Mirror of Gesture"Kathakali Dance-drama: Where Gods and Demons Come to Play"Kathakali"Indian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceMedieval Indian Literature: An AnthologyThe Oxford Companion to Indian TheatreSouth Asian Folklore: An Encyclopedia : Afghanistan, Bangladesh, India, Nepal, Pakistan, Sri LankaThe Rise of Performance Studies: Rethinking Richard Schechner's Broad SpectrumIndian Theatre: Traditions of PerformanceModern Asian Theatre and Performance 1900-2000Critical Theory and PerformanceBetween Theater and AnthropologyKathakali603847011Indian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceIndian Theatre: Traditions of PerformanceBetween Theater and AnthropologyBetween Theater and AnthropologyNambeesan Smaraka AwardsArchivedThe Cambridge Guide to TheatreRoutledge International Encyclopedia of Women: Global Women's Issues and KnowledgeThe Garland Encyclopedia of World Music: South Asia : the Indian subcontinentThe Ethos of Noh: Actors and Their Art10.2307/1145740By Means of Performance: Intercultural Studies of Theatre and Ritual10.1017/s204912550000100xReconceiving the Renaissance: A Critical ReaderPerformance TheoryListening to Theatre: The Aural Dimension of Beijing Opera10.2307/1146013Kathakali: The Art of the Non-WorldlyOn KathakaliKathakali, the dance theatreThe Kathakali Complex: Performance & StructureKathakali Dance-Drama: Where Gods and Demons Come to Play10.1093/obo/9780195399318-0071Drama and Ritual of Early Hinduism"In the Shadow of Hollywood Orientalism: Authentic East Indian Dancing"10.1080/08949460490274013Sanskrit Play Production in Ancient IndiaIndian Music: History and StructureBharata, the Nāṭyaśāstra233639306Table of Contents2238067286469807Dance In Indian Painting10.2307/32047833204783Kathakali Dance-Theatre: A Visual Narrative of Sacred Indian MimeIndian Classical Dance: The Renaissance and BeyondKathakali: an indigenous art-form of Keralaeee

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