summaryrefslogtreecommitdiffstats
path: root/src/gui/graphicsview/qsimplex_p.cpp
blob: eb8bcb8c1f465c1ccffb65bced9f6f905f10bfac (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
/****************************************************************************
**
** Copyright (C) 2011 Nokia Corporation and/or its subsidiary(-ies).
** All rights reserved.
** Contact: Nokia Corporation (qt-info@nokia.com)
**
** This file is part of the QtGui module of the Qt Toolkit.
**
** $QT_BEGIN_LICENSE:LGPL$
** GNU Lesser General Public License Usage
** This file may be used under the terms of the GNU Lesser General Public
** License version 2.1 as published by the Free Software Foundation and
** appearing in the file LICENSE.LGPL included in the packaging of this
** file. Please review the following information to ensure the GNU Lesser
** General Public License version 2.1 requirements will be met:
** http://www.gnu.org/licenses/old-licenses/lgpl-2.1.html.
**
** In addition, as a special exception, Nokia gives you certain additional
** rights. These rights are described in the Nokia Qt LGPL Exception
** version 1.1, included in the file LGPL_EXCEPTION.txt in this package.
**
** GNU General Public License Usage
** Alternatively, this file may be used under the terms of the GNU General
** Public License version 3.0 as published by the Free Software Foundation
** and appearing in the file LICENSE.GPL included in the packaging of this
** file. Please review the following information to ensure the GNU General
** Public License version 3.0 requirements will be met:
** http://www.gnu.org/copyleft/gpl.html.
**
** Other Usage
** Alternatively, this file may be used in accordance with the terms and
** conditions contained in a signed written agreement between you and Nokia.
**
**
**
**
**
** $QT_END_LICENSE$
**
****************************************************************************/

#include "qsimplex_p.h"

#include <QtCore/qset.h>
#include <QtCore/qdebug.h>

#include <stdlib.h>

QT_BEGIN_NAMESPACE

/*!
  \internal
  \class QSimplex

  The QSimplex class is a Linear Programming problem solver based on the two-phase
  simplex method.

  It takes a set of QSimplexConstraints as its restrictive constraints and an
  additional QSimplexConstraint as its objective function. Then methods to maximize
  and minimize the problem solution are provided.

  The two-phase simplex method is based on the following steps:
  First phase:
  1.a) Modify the original, complex, and possibly not feasible problem, into a new,
       easy to solve problem.
  1.b) Set as the objective of the new problem, a feasible solution for the original
       complex problem.
  1.c) Run simplex to optimize the modified problem and check whether a solution for
       the original problem exists.

  Second phase:
  2.a) Go back to the original problem with the feasibl (but not optimal) solution
       found in the first phase.
  2.b) Set the original objective.
  3.c) Run simplex to optimize the original problem towards its optimal solution.
*/

/*!
  \internal
*/
QSimplex::QSimplex() : objective(0), rows(0), columns(0), firstArtificial(0), matrix(0)
{
}

/*!
  \internal
*/
QSimplex::~QSimplex()
{
    clearDataStructures();
}

/*!
  \internal
*/
void QSimplex::clearDataStructures()
{
    if (matrix == 0)
        return;

    // Matrix
    rows = 0;
    columns = 0;
    firstArtificial = 0;
    free(matrix);
    matrix = 0;

    // Constraints
    for (int i = 0; i < constraints.size(); ++i) {
        delete constraints[i]->helper.first;
        delete constraints[i]->artificial;
        delete constraints[i];
    }
    constraints.clear();

    // Other
    variables.clear();
    objective = 0;
}

/*!
  \internal
  Sets the new constraints in the simplex solver and returns whether the problem
  is feasible.

  This method sets the new constraints, normalizes them, creates the simplex matrix
  and runs the first simplex phase.
*/
bool QSimplex::setConstraints(const QList<QSimplexConstraint *> newConstraints)
{
    ////////////////////////////
    // Reset to initial state //
    ////////////////////////////
    clearDataStructures();

    if (newConstraints.isEmpty())
        return true;    // we are ok with no constraints

    // Make deep copy of constraints. We need this copy because we may change
    // them in the simplification method.
    for (int i = 0; i < newConstraints.size(); ++i) {
        QSimplexConstraint *c = new QSimplexConstraint;
        c->constant = newConstraints[i]->constant;
        c->ratio = newConstraints[i]->ratio;
        c->variables = newConstraints[i]->variables;
        constraints << c;
    }

    // Remove constraints of type Var == K and replace them for their value.
    if (!simplifyConstraints(&constraints)) {
        qWarning() << "QSimplex: No feasible solution!";
        clearDataStructures();
        return false;
    }

    ///////////////////////////////////////
    // Prepare variables and constraints //
    ///////////////////////////////////////

    // Set Variables direct mapping.
    // "variables" is a list that provides a stable, indexed list of all variables
    // used in this problem.
    QSet<QSimplexVariable *> variablesSet;
    for (int i = 0; i < constraints.size(); ++i)
        variablesSet += \
            QSet<QSimplexVariable *>::fromList(constraints[i]->variables.keys());
    variables = variablesSet.toList();

    // Set Variables reverse mapping
    // We also need to be able to find the index for a given variable, to do that
    // we store in each variable its index.
    for (int i = 0; i < variables.size(); ++i) {
        // The variable "0" goes at the column "1", etc...
        variables[i]->index = i + 1;
    }

    // Normalize Constraints
    // In this step, we prepare the constraints in two ways:
    // Firstly, we modify all constraints of type "LessOrEqual" or "MoreOrEqual"
    // by the adding slack or surplus variables and making them "Equal" constraints.
    // Secondly, we need every single constraint to have a direct, easy feasible
    // solution. Constraints that have slack variables are already easy to solve,
    // to all the others we add artificial variables.
    //
    // At the end we modify the constraints as follows:
    //  - LessOrEqual: SLACK variable is added.
    //  - Equal: ARTIFICIAL variable is added.
    //  - More or Equal: ARTIFICIAL and SURPLUS variables are added.
    int variableIndex = variables.size();
    QList <QSimplexVariable *> artificialList;

    for (int i = 0; i < constraints.size(); ++i) {
        QSimplexVariable *slack;
        QSimplexVariable *surplus;
        QSimplexVariable *artificial;

        Q_ASSERT(constraints[i]->helper.first == 0);
        Q_ASSERT(constraints[i]->artificial == 0);

        switch(constraints[i]->ratio) {
        case QSimplexConstraint::LessOrEqual:
            slack = new QSimplexVariable;
            slack->index = ++variableIndex;
            constraints[i]->helper.first = slack;
            constraints[i]->helper.second = 1.0;
            break;
        case QSimplexConstraint::MoreOrEqual:
            surplus = new QSimplexVariable;
            surplus->index = ++variableIndex;
            constraints[i]->helper.first = surplus;
            constraints[i]->helper.second = -1.0;
            // fall through
        case QSimplexConstraint::Equal:
            artificial = new QSimplexVariable;
            constraints[i]->artificial = artificial;
            artificialList += constraints[i]->artificial;
            break;
        }
    }

    // All original, slack and surplus have already had its index set
    // at this point. We now set the index of the artificial variables
    // as to ensure they are at the end of the variable list and therefore
    // can be easily removed at the end of this method.
    firstArtificial = variableIndex + 1;
    for (int i = 0; i < artificialList.size(); ++i)
        artificialList[i]->index = ++variableIndex;
    artificialList.clear();

    /////////////////////////////
    // Fill the Simplex matrix //
    /////////////////////////////

    // One for each variable plus the Basic and BFS columns (first and last)
    columns = variableIndex + 2;
    // One for each constraint plus the objective function
    rows = constraints.size() + 1;

    matrix = (qreal *)malloc(sizeof(qreal) * columns * rows);
    if (!matrix) {
        qWarning() << "QSimplex: Unable to allocate memory!";
        return false;
    }
    for (int i = columns * rows - 1; i >= 0; --i)
        matrix[i] = 0.0;

    // Fill Matrix
    for (int i = 1; i <= constraints.size(); ++i) {
        QSimplexConstraint *c = constraints[i - 1];

        if (c->artificial) {
            // Will use artificial basic variable
            setValueAt(i, 0, c->artificial->index);
            setValueAt(i, c->artificial->index, 1.0);

            if (c->helper.second != 0.0) {
                // Surplus variable
                setValueAt(i, c->helper.first->index, c->helper.second);
            }
        } else {
            // Slack is used as the basic variable
            Q_ASSERT(c->helper.second == 1.0);
            setValueAt(i, 0, c->helper.first->index);
            setValueAt(i, c->helper.first->index, 1.0);
        }

        QHash<QSimplexVariable *, qreal>::const_iterator iter;
        for (iter = c->variables.constBegin();
             iter != c->variables.constEnd();
             ++iter) {
            setValueAt(i, iter.key()->index, iter.value());
        }

        setValueAt(i, columns - 1, c->constant);
    }

    // Set objective for the first-phase Simplex.
    // Z = -1 * sum_of_artificial_vars
    for (int j = firstArtificial; j < columns - 1; ++j)
        setValueAt(0, j, 1.0);

    // Maximize our objective (artificial vars go to zero)
    solveMaxHelper();

    // If there is a solution where the sum of all artificial
    // variables is zero, then all of them can be removed and yet
    // we will have a feasible (but not optimal) solution for the
    // original problem.
    // Otherwise, we clean up our structures and report there is
    // no feasible solution.
    if ((valueAt(0, columns - 1) != 0.0) && (qAbs(valueAt(0, columns - 1)) > 0.00001)) {
        qWarning() << "QSimplex: No feasible solution!";
        clearDataStructures();
        return false;
    }

    // Remove artificial variables. We already have a feasible
    // solution for the first problem, thus we don't need them
    // anymore.
    clearColumns(firstArtificial, columns - 2);

    return true;
}

/*!
  \internal

  Run simplex on the current matrix with the current objective.

  This is the iterative method. The matrix lines are combined
  as to modify the variable values towards the best solution possible.
  The method returns when the matrix is in the optimal state.
*/
void QSimplex::solveMaxHelper()
{
    reducedRowEchelon();
    while (iterate()) ;
}

/*!
  \internal
*/
void QSimplex::setObjective(QSimplexConstraint *newObjective)
{
    objective = newObjective;
}

/*!
  \internal
*/
void QSimplex::clearRow(int rowIndex)
{
    qreal *item = matrix + rowIndex * columns;
    for (int i = 0; i < columns; ++i)
        item[i] = 0.0;
}

/*!
  \internal
*/
void QSimplex::clearColumns(int first, int last)
{
    for (int i = 0; i < rows; ++i) {
        qreal *row = matrix + i * columns;
        for (int j = first; j <= last; ++j)
            row[j] = 0.0;
    }
}

/*!
  \internal
*/
void QSimplex::dumpMatrix()
{
    qDebug("---- Simplex Matrix ----\n");

    QString str(QLatin1String("       "));
    for (int j = 0; j < columns; ++j)
        str += QString::fromAscii("  <%1 >").arg(j, 2);
    qDebug("%s", qPrintable(str));
    for (int i = 0; i < rows; ++i) {
        str = QString::fromAscii("Row %1:").arg(i, 2);

        qreal *row = matrix + i * columns;
        for (int j = 0; j < columns; ++j)
            str += QString::fromAscii("%1").arg(row[j], 7, 'f', 2);
        qDebug("%s", qPrintable(str));
    }
    qDebug("------------------------\n");
}

/*!
  \internal
*/
void QSimplex::combineRows(int toIndex, int fromIndex, qreal factor)
{
    if (!factor)
        return;

    qreal *from = matrix + fromIndex * columns;
    qreal *to = matrix + toIndex * columns;

    for (int j = 1; j < columns; ++j) {
        qreal value = from[j];

        // skip to[j] = to[j] + factor*0.0
        if (value == 0.0)
            continue;

        to[j] += factor * value;

        // ### Avoid Numerical errors
        if (qAbs(to[j]) < 0.0000000001)
            to[j] = 0.0;
    }
}

/*!
  \internal
*/
int QSimplex::findPivotColumn()
{
    qreal min = 0;
    int minIndex = -1;

    for (int j = 0; j < columns-1; ++j) {
        if (valueAt(0, j) < min) {
            min = valueAt(0, j);
            minIndex = j;
        }
    }

    return minIndex;
}

/*!
  \internal

  For a given pivot column, find the pivot row. That is, the row with the
  minimum associated "quotient" where:

  - quotient is the division of the value in the last column by the value
    in the pivot column.
  - rows with value less or equal to zero are ignored
  - if two rows have the same quotient, lines are chosen based on the
    highest variable index (value in the first column)

  The last condition avoids a bug where artificial variables would be
  left behind for the second-phase simplex, and with 'good'
  constraints would be removed before it, what would lead to incorrect
  results.
*/
int QSimplex::pivotRowForColumn(int column)
{
    qreal min = qreal(999999999999.0); // ###
    int minIndex = -1;

    for (int i = 1; i < rows; ++i) {
        qreal divisor = valueAt(i, column);
        if (divisor <= 0)
            continue;

        qreal quotient = valueAt(i, columns - 1) / divisor;
        if (quotient < min) {
            min = quotient;
            minIndex = i;
        } else if ((quotient == min) && (valueAt(i, 0) > valueAt(minIndex, 0))) {
            minIndex = i;
        }
    }

    return minIndex;
}

/*!
  \internal
*/
void QSimplex::reducedRowEchelon()
{
    for (int i = 1; i < rows; ++i) {
        int factorInObjectiveRow = valueAt(i, 0);
        combineRows(0, i, -1 * valueAt(0, factorInObjectiveRow));
    }
}

/*!
  \internal

  Does one iteration towards a better solution for the problem.
  See 'solveMaxHelper'.
*/
bool QSimplex::iterate()
{
    // Find Pivot column
    int pivotColumn = findPivotColumn();
    if (pivotColumn == -1)
        return false;

    // Find Pivot row for column
    int pivotRow = pivotRowForColumn(pivotColumn);
    if (pivotRow == -1) {
        qWarning() << "QSimplex: Unbounded problem!";
        return false;
    }

    // Normalize Pivot Row
    qreal pivot = valueAt(pivotRow, pivotColumn);
    if (pivot != 1.0)
        combineRows(pivotRow, pivotRow, (1.0 - pivot) / pivot);

    // Update other rows
    for (int row=0; row < rows; ++row) {
        if (row == pivotRow)
            continue;

        combineRows(row, pivotRow, -1 * valueAt(row, pivotColumn));
    }

    // Update first column
    setValueAt(pivotRow, 0, pivotColumn);

    //    dumpMatrix();
    //    qDebug("------------ end of iteration --------------\n");
    return true;
}

/*!
  \internal

  Both solveMin and solveMax are interfaces to this method.

  The enum solverFactor admits 2 values: Minimum (-1) and Maximum (+1).

  This method sets the original objective and runs the second phase
  Simplex to obtain the optimal solution for the problem. As the internal
  simplex solver is only able to _maximize_ objectives, we handle the
  minimization case by inverting the original objective and then
  maximizing it.
*/
qreal QSimplex::solver(solverFactor factor)
{
    // Remove old objective
    clearRow(0);

    // Set new objective in the first row of the simplex matrix
    qreal resultOffset = 0;
    QHash<QSimplexVariable *, qreal>::const_iterator iter;
    for (iter = objective->variables.constBegin();
         iter != objective->variables.constEnd();
         ++iter) {

        // Check if the variable was removed in the simplification process.
        // If so, we save its offset to the objective function and skip adding
        // it to the matrix.
        if (iter.key()->index == -1) {
            resultOffset += iter.value() * iter.key()->result;
            continue;
        }

        setValueAt(0, iter.key()->index, -1 * factor * iter.value());
    }

    solveMaxHelper();
    collectResults();

#ifdef QT_DEBUG
    for (int i = 0; i < constraints.size(); ++i) {
        Q_ASSERT(constraints[i]->isSatisfied());
    }
#endif

    // Return the value calculated by the simplex plus the value of the
    // fixed variables.
    return (factor * valueAt(0, columns - 1)) + resultOffset;
}

/*!
  \internal
  Minimize the original objective.
*/
qreal QSimplex::solveMin()
{
    return solver(Minimum);
}

/*!
  \internal
  Maximize the original objective.
*/
qreal QSimplex::solveMax()
{
    return solver(Maximum);
}

/*!
  \internal

  Reads results from the simplified matrix and saves them in the
  "result" member of each QSimplexVariable.
*/
void QSimplex::collectResults()
{
    // All variables are zero unless overridden below.

    // ### Is this really needed? Is there any chance that an
    // important variable remains as non-basic at the end of simplex?
    for (int i = 0; i < variables.size(); ++i)
        variables[i]->result = 0;

    // Basic variables
    // Update the variable indicated in the first column with the value
    // in the last column.
    for (int i = 1; i < rows; ++i) {
        int index = valueAt(i, 0) - 1;
        if (index < variables.size())
            variables[index]->result = valueAt(i, columns - 1);
    }
}

/*!
  \internal

  Looks for single-valued variables and remove them from the constraints list.
*/
bool QSimplex::simplifyConstraints(QList<QSimplexConstraint *> *constraints)
{
    QHash<QSimplexVariable *, qreal> results;   // List of single-valued variables
    bool modified = true;                       // Any chance more optimization exists?

    while (modified) {
        modified = false;

        // For all constraints
        QList<QSimplexConstraint *>::iterator iter = constraints->begin();
        while (iter != constraints->end()) {
            QSimplexConstraint *c = *iter;
            if ((c->ratio == QSimplexConstraint::Equal) && (c->variables.count() == 1)) {
                // Check whether this is a constraint of type Var == K
                // If so, save its value to "results".
                QSimplexVariable *variable = c->variables.constBegin().key();
                qreal result = c->constant / c->variables.value(variable);

                results.insert(variable, result);
                variable->result = result;
                variable->index = -1;
                modified = true;

            }

            // Replace known values among their variables
            QHash<QSimplexVariable *, qreal>::const_iterator r;
            for (r = results.constBegin(); r != results.constEnd(); ++r) {
                if (c->variables.contains(r.key())) {
                    c->constant -= r.value() * c->variables.take(r.key());
                    modified = true;
                }
            }

            // Keep it normalized
            if (c->constant < 0)
                c->invert();

            if (c->variables.isEmpty()) {
                // If constraint became empty due to substitution, delete it.
                if (c->isSatisfied() == false)
                    // We must ensure that the constraint soon to be deleted would not
                    // make the problem unfeasible if left behind. If that's the case,
                    // we return false so the simplex solver can properly report that.
                    return false;

                delete c;
                iter = constraints->erase(iter);
            } else {
                ++iter;
            }
        }
    }

    return true;
}

void QSimplexConstraint::invert()
{
    constant = -constant;
    ratio = Ratio(2 - ratio);

    QHash<QSimplexVariable *, qreal>::iterator iter;
    for (iter = variables.begin(); iter != variables.end(); ++iter) {
        iter.value() = -iter.value();
    }
}

QT_END_NAMESPACE