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Improve the quadratic-constraint.ipynb
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‎examples/quadratic-constraints.ipynb‎

Lines changed: 128 additions & 54 deletions
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "imports",
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"metadata": {
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"ExecuteTime": {
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"end_time": "2025-11-26T10:12:07.434614Z",
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"start_time": "2025-11-26T10:12:07.424509Z"
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"end_time": "2025-12-03T14:04:49.055855Z",
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"start_time": "2025-12-03T14:04:45.460505Z"
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}
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},
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"outputs": [],
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"source": [
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"import pandas as pd\n",
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"\n",
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"import linopy"
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],
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"outputs": [],
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"execution_count": null
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]
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},
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{
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"cell_type": "markdown",
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "model-setup",
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"metadata": {
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"ExecuteTime": {
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"end_time": "2025-11-26T10:12:07.460556Z",
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"start_time": "2025-11-26T10:12:07.442789Z"
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"end_time": "2025-12-03T14:04:49.840053Z",
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"start_time": "2025-12-03T14:04:49.669343Z"
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}
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},
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"outputs": [],
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"source": [
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"m = linopy.Model()\n",
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"\n",
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"time = pd.Index(range(5), name=\"time\")\n",
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"\n",
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"x = m.add_variables(lower=0, coords=[time], name=\"x\")\n",
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"y = m.add_variables(lower=0, coords=[time], name=\"y\")"
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],
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"outputs": [],
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"execution_count": null
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]
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},
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{
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"cell_type": "markdown",
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "linear-constraint",
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"metadata": {
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"ExecuteTime": {
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"end_time": "2025-11-26T10:12:07.487172Z",
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"start_time": "2025-11-26T10:12:07.464747Z"
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"end_time": "2025-12-03T14:04:50.203648Z",
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"start_time": "2025-12-03T14:04:50.079589Z"
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}
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},
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"outputs": [],
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"source": [
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"budget = pd.Series([8, 9, 10, 11, 12], index=time)\n",
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"\n",
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"m.add_constraints(x + y <= budget, name=\"budget\")"
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],
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"outputs": [],
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"execution_count": null
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]
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},
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{
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"cell_type": "markdown",
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "quad-constraint",
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"metadata": {
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"ExecuteTime": {
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"end_time": "2025-11-26T10:12:07.540835Z",
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"start_time": "2025-11-26T10:12:07.506741Z"
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"end_time": "2025-12-03T14:04:50.639153Z",
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"start_time": "2025-12-03T14:04:50.544579Z"
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}
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},
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"outputs": [],
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"source": [
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"risk_limit = pd.Series([20, 25, 30, 35, 40], index=time)\n",
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"\n",
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"m.add_quadratic_constraints(x**2 + y**2, \"<=\", risk_limit, name=\"risk\")"
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],
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"outputs": [],
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"execution_count": null
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]
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},
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{
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"cell_type": "markdown",
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "objective",
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"metadata": {
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"ExecuteTime": {
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"end_time": "2025-11-26T10:12:07.554745Z",
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"start_time": "2025-11-26T10:12:07.545752Z"
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"end_time": "2025-12-03T14:04:51.366506Z",
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"start_time": "2025-12-03T14:04:51.276869Z"
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}
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},
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"outputs": [],
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"source": [
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"m.add_objective((x + 2 * y).sum(), sense=\"max\")\n",
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"m"
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],
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"outputs": [],
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"execution_count": null
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]
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},
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{
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"cell_type": "markdown",
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "solve",
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"metadata": {
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"ExecuteTime": {
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"end_time": "2025-11-26T10:12:07.741963Z",
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"start_time": "2025-11-26T10:12:07.568872Z"
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"end_time": "2025-12-03T14:04:52.311740Z",
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"start_time": "2025-12-03T14:04:51.994541Z"
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}
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},
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"outputs": [],
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"source": [
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"m.solve(solver_name=\"gurobi\", QCPDual=1)"
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],
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"outputs": [],
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"execution_count": null
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]
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},
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{
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"cell_type": "markdown",
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "results",
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"metadata": {
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"ExecuteTime": {
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"end_time": "2025-11-26T10:12:07.833812Z",
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"start_time": "2025-11-26T10:12:07.752355Z"
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"end_time": "2025-12-03T14:04:57.706433Z",
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"start_time": "2025-12-03T14:04:56.543039Z"
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}
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},
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"outputs": [],
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"source": [
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"m.solution.to_dataframe().plot(kind=\"bar\", ylabel=\"Optimal Value\", rot=0);"
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],
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"outputs": [],
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"execution_count": null
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]
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},
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{
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"cell_type": "markdown",
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "qc-container",
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"metadata": {
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"ExecuteTime": {
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"end_time": "2025-11-26T10:12:07.841509Z",
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"start_time": "2025-11-26T10:12:07.837808Z"
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"end_time": "2025-12-03T14:04:58.261183Z",
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"start_time": "2025-12-03T14:04:58.251877Z"
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}
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},
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"outputs": [],
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"source": [
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"m.quadratic_constraints"
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],
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"outputs": [],
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"execution_count": null
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "qc-single",
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"metadata": {
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"ExecuteTime": {
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"end_time": "2025-11-26T10:12:07.854545Z",
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"start_time": "2025-11-26T10:12:07.847875Z"
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"end_time": "2025-12-03T14:04:58.556810Z",
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"start_time": "2025-12-03T14:04:58.521816Z"
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}
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},
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"outputs": [],
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"source": [
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"m.quadratic_constraints[\"risk\"]"
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],
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"outputs": [],
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"execution_count": null
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]
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},
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"cell_type": "markdown",
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "qc-duals",
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"metadata": {
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"ExecuteTime": {
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"end_time": "2025-11-26T10:12:07.887976Z",
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"start_time": "2025-11-26T10:12:07.876458Z"
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"end_time": "2025-12-03T14:04:58.728502Z",
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"start_time": "2025-12-03T14:04:58.714225Z"
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}
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},
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"outputs": [],
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"source": [
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"m.quadratic_constraints[\"risk\"].dual"
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],
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"outputs": [],
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"execution_count": null
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]
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},
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{
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"cell_type": "markdown",
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "bilinear",
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"metadata": {
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"ExecuteTime": {
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"end_time": "2025-11-26T10:12:07.998397Z",
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"start_time": "2025-11-26T10:12:07.922217Z"
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"end_time": "2025-12-03T14:04:59.651299Z",
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"start_time": "2025-12-03T14:04:59.297236Z"
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}
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},
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"outputs": [],
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"source": [
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"m2 = linopy.Model()\n",
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"\n",
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"m2.solve(solver_name=\"gurobi\")\n",
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"\n",
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"print(f\"x = {float(x.solution):.2f}, y = {float(y.solution):.2f}\")"
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],
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]
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},
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{
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"cell_type": "markdown",
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"id": "w5c0l9z5efi",
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"metadata": {},
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"source": "## Mixed Linear and Quadratic Terms\n\nQuadratic constraints can combine both quadratic and linear terms in the same constraint. For example, $x^2 + 2x + y \\leq 10$:"
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "gtgvi4i0mzl",
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"metadata": {
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"ExecuteTime": {
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"end_time": "2025-12-03T14:05:02.498083Z",
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"start_time": "2025-12-03T14:05:02.254357Z"
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}
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},
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"outputs": [],
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"source": [
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"m3 = linopy.Model()\n",
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"\n",
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"x = m3.add_variables(lower=0, name=\"x\")\n",
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"y = m3.add_variables(lower=0, name=\"y\")\n",
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"\n",
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"# Mixed constraint: x² + 2x + y <= 10\n",
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"m3.add_quadratic_constraints(x**2 + 2 * x + y, \"<=\", 10, name=\"mixed\")\n",
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"\n",
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"m3.add_objective(x + y, sense=\"max\")\n",
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"\n",
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"m3.solve(solver_name=\"gurobi\")\n",
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"\n",
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"print(f\"x = {float(x.solution):.2f}, y = {float(y.solution):.2f}\")"
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]
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},
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{
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"cell_type": "markdown",
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"id": "5gisroyuys4",
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"metadata": {},
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"source": "## Equality Constraints\n\nQuadratic equality constraints use `\"==\"`. This example constrains a point to lie exactly on a circle:"
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "x7tzlfma7fq",
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"metadata": {
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"ExecuteTime": {
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"end_time": "2025-12-03T14:05:05.330521Z",
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"start_time": "2025-12-03T14:05:05.202101Z"
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}
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},
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"outputs": [],
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"execution_count": null
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"source": [
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"m4 = linopy.Model()\n",
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"\n",
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"x = m4.add_variables(lower=-5, upper=5, name=\"x\")\n",
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"y = m4.add_variables(lower=-5, upper=5, name=\"y\")\n",
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"\n",
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"# Point must lie on circle of radius 2\n",
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"m4.add_quadratic_constraints(x**2 + y**2, \"==\", 4, name=\"circle\")\n",
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"\n",
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"# Maximize x + y (find point on circle furthest in direction (1,1))\n",
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"m4.add_objective(x + y, sense=\"max\")\n",
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"\n",
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"m4.solve(solver_name=\"gurobi\")\n",
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"\n",
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"print(f\"x = {float(x.solution):.4f}, y = {float(y.solution):.4f}\")\n",
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"print(f\"x² + y² = {float(x.solution) ** 2 + float(y.solution) ** 2:.4f}\")"
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]
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},
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{
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"cell_type": "markdown",
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"id": "ru3y88un3an",
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"metadata": {},
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"source": "## Convexity Considerations\n\nQuadratic constraints have important convexity properties that affect which solvers can handle them:\n\n- **Convex** (most solvers): $x^T Q x + a^T x \\leq b$ where $Q$ is positive semidefinite (e.g., sum of squares like $x^2 + y^2$)\n- **Nonconvex** (requires specialized solvers like Gurobi):\n - $x^T Q x + a^T x \\geq b$ (greater-than with positive semidefinite Q)\n - $x^T Q x + a^T x = b$ (equality constraints)\n - Bilinear terms like $xy$\n\nConvex quadratic constraints define a convex feasible region (like the interior of an ellipse), while nonconvex constraints can create disconnected or non-convex regions. Solvers like Gurobi use spatial branch-and-bound to handle nonconvex cases."
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}
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],
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"metadata": {

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