Mixing
Binary Polymatroid Optimization Problem
$begingroup$
Let $mathcal{N}$ denote the finite set ${1, 2, ldots, n}$, and let $mathcal{S}_j$ denote the set ${1, 2, ldots, j}$; let $fcolon mathcal{N} to mathbb{N}$ be nondecreasing, submodular and with $f(emptyset) = 0$, and let $c_1, c_2, ldots, c_n$ be an nonnegative integer-valued cost vector satisfying $c_i geq c_j$ when $i leq j$. We seek an assignment vector $x^* in {0, 1}^n$ maximizing
begin{equation}
sum_{i = 1}^n c_ix_i text{ under the constraint } sum_{i in mathcal{S}} x_i leq f(mathcal{S}) text{ for all } mathcal{S} subseteq mathcal{N}.
end{equation}
Note that if $x_i geq 0, forall i in mathcal{N}$ (as opposed to binary), then the optimal assignment is given by $x_i = f(mathcal{S}_i) - f(mathcal{S}_{i - 1})$ (cf. Optimization over Integers by Bertsimas & Weismantel). My guess is that
begin{equation}
x^* = begin{cases}
1 & text{ if }: f(mathcal{S}_i) - f(mathcal{S}_{i - 1}) > 0\
0 & text{ else.}
end{cases}
end{equation}
To establish the optimality of my proposed solution, I would like to construct a dual-feasible solution of the same cost/value, i.e., to find a nonnegative (componentwise) vector $y_mathcal{S}, mathcal{S} subseteq mathcal{N},$ and a nonnegative vector $tilde{y}_i, i in mathcal{N},$ such that
begin{equation}
sum_{mathcal{S}: i in mathcal{S}} y_mathcal{S} + tilde{y}_i geq c_i, forall i in mathcal{N} text{ (dual feasibility)},
end{equation}
and such that
begin{equation}
sum_{mathcal{S} subseteq mathcal{N}} y_mathcal{S}f(mathcal{S}) + sum_{i = 1}^n tilde{y}_i = sum_{i = 1}^n c_ix^*_i text { (optimality via weak duality)}.
end{equation}
Note again that without restricting $x$ to be binary, a suitable dual vector is $y_mathcal{S} = c_i - c_{i + 1}$ if $mathcal{S} = mathcal{S}_i$, and $y_mathcal{S} = 0$ else ($tilde{y}_i = 0, forall i in mathcal{N}$). I appreciate any help towards finding a suitable dual vector to my proposed solution $x^*$ (or any other argument as to why $x^*$ is the optimal assignment).
linear-algebra duality-theorems discrete-optimization
asked Jan 17 at 23:45
user480881user480881
1118
$endgroup$
add a comment |
$begingroup$
Let $mathcal{N}$ denote the finite set ${1, 2, ldots, n}$, and let $mathcal{S}_j$ denote the set ${1, 2, ldots, j}$; let $fcolon mathcal{N} to mathbb{N}$ be nondecreasing, submodular and with $f(emptyset) = 0$, and let $c_1, c_2, ldots, c_n$ be an nonnegative integer-valued cost vector satisfying $c_i geq c_j$ when $i leq j$. We seek an assignment vector $x^* in {0, 1}^n$ maximizing
begin{equation}
sum_{i = 1}^n c_ix_i text{ under the constraint } sum_{i in mathcal{S}} x_i leq f(mathcal{S}) text{ for all } mathcal{S} subseteq mathcal{N}.
end{equation}
Note that if $x_i geq 0, forall i in mathcal{N}$ (as opposed to binary), then the optimal assignment is given by $x_i = f(mathcal{S}_i) - f(mathcal{S}_{i - 1})$ (cf. Optimization over Integers by Bertsimas & Weismantel). My guess is that
begin{equation}
x^* = begin{cases}
1 & text{ if }: f(mathcal{S}_i) - f(mathcal{S}_{i - 1}) > 0\
0 & text{ else.}
end{cases}
end{equation}
To establish the optimality of my proposed solution, I would like to construct a dual-feasible solution of the same cost/value, i.e., to find a nonnegative (componentwise) vector $y_mathcal{S}, mathcal{S} subseteq mathcal{N},$ and a nonnegative vector $tilde{y}_i, i in mathcal{N},$ such that
begin{equation}
sum_{mathcal{S}: i in mathcal{S}} y_mathcal{S} + tilde{y}_i geq c_i, forall i in mathcal{N} text{ (dual feasibility)},
end{equation}
and such that
begin{equation}
sum_{mathcal{S} subseteq mathcal{N}} y_mathcal{S}f(mathcal{S}) + sum_{i = 1}^n tilde{y}_i = sum_{i = 1}^n c_ix^*_i text { (optimality via weak duality)}.
end{equation}
Note again that without restricting $x$ to be binary, a suitable dual vector is $y_mathcal{S} = c_i - c_{i + 1}$ if $mathcal{S} = mathcal{S}_i$, and $y_mathcal{S} = 0$ else ($tilde{y}_i = 0, forall i in mathcal{N}$). I appreciate any help towards finding a suitable dual vector to my proposed solution $x^*$ (or any other argument as to why $x^*$ is the optimal assignment).
linear-algebra duality-theorems discrete-optimization
asked Jan 17 at 23:45
user480881user480881
1118
$endgroup$
add a comment |
$begingroup$
Let $mathcal{N}$ denote the finite set ${1, 2, ldots, n}$, and let $mathcal{S}_j$ denote the set ${1, 2, ldots, j}$; let $fcolon mathcal{N} to mathbb{N}$ be nondecreasing, submodular and with $f(emptyset) = 0$, and let $c_1, c_2, ldots, c_n$ be an nonnegative integer-valued cost vector satisfying $c_i geq c_j$ when $i leq j$. We seek an assignment vector $x^* in {0, 1}^n$ maximizing
begin{equation}
sum_{i = 1}^n c_ix_i text{ under the constraint } sum_{i in mathcal{S}} x_i leq f(mathcal{S}) text{ for all } mathcal{S} subseteq mathcal{N}.
end{equation}
Note that if $x_i geq 0, forall i in mathcal{N}$ (as opposed to binary), then the optimal assignment is given by $x_i = f(mathcal{S}_i) - f(mathcal{S}_{i - 1})$ (cf. Optimization over Integers by Bertsimas & Weismantel). My guess is that
begin{equation}
x^* = begin{cases}
1 & text{ if }: f(mathcal{S}_i) - f(mathcal{S}_{i - 1}) > 0\
0 & text{ else.}
end{cases}
end{equation}
To establish the optimality of my proposed solution, I would like to construct a dual-feasible solution of the same cost/value, i.e., to find a nonnegative (componentwise) vector $y_mathcal{S}, mathcal{S} subseteq mathcal{N},$ and a nonnegative vector $tilde{y}_i, i in mathcal{N},$ such that
begin{equation}
sum_{mathcal{S}: i in mathcal{S}} y_mathcal{S} + tilde{y}_i geq c_i, forall i in mathcal{N} text{ (dual feasibility)},
end{equation}
and such that
begin{equation}
sum_{mathcal{S} subseteq mathcal{N}} y_mathcal{S}f(mathcal{S}) + sum_{i = 1}^n tilde{y}_i = sum_{i = 1}^n c_ix^*_i text { (optimality via weak duality)}.
end{equation}
Note again that without restricting $x$ to be binary, a suitable dual vector is $y_mathcal{S} = c_i - c_{i + 1}$ if $mathcal{S} = mathcal{S}_i$, and $y_mathcal{S} = 0$ else ($tilde{y}_i = 0, forall i in mathcal{N}$). I appreciate any help towards finding a suitable dual vector to my proposed solution $x^*$ (or any other argument as to why $x^*$ is the optimal assignment).
linear-algebra duality-theorems discrete-optimization
asked Jan 17 at 23:45
user480881user480881
1118
$endgroup$
Let $mathcal{N}$ denote the finite set ${1, 2, ldots, n}$, and let $mathcal{S}_j$ denote the set ${1, 2, ldots, j}$; let $fcolon mathcal{N} to mathbb{N}$ be nondecreasing, submodular and with $f(emptyset) = 0$, and let $c_1, c_2, ldots, c_n$ be an nonnegative integer-valued cost vector satisfying $c_i geq c_j$ when $i leq j$. We seek an assignment vector $x^* in {0, 1}^n$ maximizing
begin{equation}
sum_{i = 1}^n c_ix_i text{ under the constraint } sum_{i in mathcal{S}} x_i leq f(mathcal{S}) text{ for all } mathcal{S} subseteq mathcal{N}.
end{equation}
Note that if $x_i geq 0, forall i in mathcal{N}$ (as opposed to binary), then the optimal assignment is given by $x_i = f(mathcal{S}_i) - f(mathcal{S}_{i - 1})$ (cf. Optimization over Integers by Bertsimas & Weismantel). My guess is that
begin{equation}
x^* = begin{cases}
1 & text{ if }: f(mathcal{S}_i) - f(mathcal{S}_{i - 1}) > 0\
0 & text{ else.}
end{cases}
end{equation}
To establish the optimality of my proposed solution, I would like to construct a dual-feasible solution of the same cost/value, i.e., to find a nonnegative (componentwise) vector $y_mathcal{S}, mathcal{S} subseteq mathcal{N},$ and a nonnegative vector $tilde{y}_i, i in mathcal{N},$ such that
begin{equation}
sum_{mathcal{S}: i in mathcal{S}} y_mathcal{S} + tilde{y}_i geq c_i, forall i in mathcal{N} text{ (dual feasibility)},
end{equation}
and such that
begin{equation}
sum_{mathcal{S} subseteq mathcal{N}} y_mathcal{S}f(mathcal{S}) + sum_{i = 1}^n tilde{y}_i = sum_{i = 1}^n c_ix^*_i text { (optimality via weak duality)}.
end{equation}
Note again that without restricting $x$ to be binary, a suitable dual vector is $y_mathcal{S} = c_i - c_{i + 1}$ if $mathcal{S} = mathcal{S}_i$, and $y_mathcal{S} = 0$ else ($tilde{y}_i = 0, forall i in mathcal{N}$). I appreciate any help towards finding a suitable dual vector to my proposed solution $x^*$ (or any other argument as to why $x^*$ is the optimal assignment).
linear-algebra duality-theorems discrete-optimization
linear-algebra duality-theorems discrete-optimization
asked Jan 17 at 23:45
user480881user480881
1118
asked Jan 17 at 23:45
user480881user480881
1118
asked Jan 17 at 23:45
user480881user480881
1118
asked Jan 17 at 23:45
user480881user480881
1118
asked Jan 17 at 23:45
user480881user480881
1118
1118
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3077677%2fbinary-polymatroid-optimization-problem%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3077677%2fbinary-polymatroid-optimization-problem%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
