<div dir="ltr"><div dir="ltr"><div dir="ltr">Hi Pietro,<div><br></div><div>First of all, thank you very much!</div><div><br></div><div>All variables are nonnegative integers. </div><div>After three days, the output of Couene is shown below:</div><div><div><img src="cid:ii_kblgmvz70" alt="Screen Shot 2020-06-18 at 8.12.41 PM.png" width="500" height="146" style="outline:0px;margin-right:0px"><br></div></div><div><br></div><div>I agree with this:</div><div>'''</div><span style="color:rgb(80,0,80)"><div><span style="font-family:Verdana;font-size:12px">Probably the quickest solution is the following: given that all w variables are integer, there are only 22 values for the denominator that's in common to all expressions (unless I'm missing something from the problem). That means you can solve 22 problems (all of them MIPs, i.e. linear with integrality constraints) and choose the best solution out of these 22. Given the size of the problem, Cbc should solve all of these 22 problems quite fast (definitely less than three days).</span><br></div></span><div><span style="font-family:Verdana;font-size:12px">'''</span></div><div><span style="font-family:Verdana;font-size:12px"><br></span></div><div><font face="Verdana"><span style="font-size:12px">However, the previous example I showed</span></font><font face="Verdana"><span style="font-size:12px"> is one easy special case. A more general case would have different </span></font><span style="font-family:Verdana;font-size:12px">denominators.</span></div><div><span style="font-family:Verdana;font-size:12px">For a tiny example,</span></div><div><span style="font-family:Verdana;font-size:12px">Objective function: </span></div><div><span style="font-family:Verdana;font-size:12px">Minimize</span></div><div>abs((w0 + w1 + w2)<b><font size="4">/</font></b>(w0 + w1 + w2 + w3 + w4 + w5) - 0.7413793103448276)<span style="font-family:Verdana;font-size:12px"><br></span></div><div>+ abs((w0 + w1 + w2)<b><font size="4">/</font></b>(w0 + w1 + w2 + w6) - 0.54455432443)</div><div>+ abs((w2 + w3)<b><font size="4">/</font></b>(w0 + w2 + w3 + w4 + w7) - 0.345453456)<br></div><div></div><div><div><br></div><div>Constraints:</div><div>(1) all variables are nonnegative integers</div></div><div><br></div><div>(2) 50 <= w0 + w1 + w2 + w3 + w4 + w5</div><div><br></div><div>(3) w0 + w1 + w2 + w3 + w4 + w5 <= 60</div><div><br></div><div>(4) 30 <= w0 + w1 + w2 + w6</div><div><br></div><div>(5) w0 + w1 + w2 + w6 <= 40</div><div><br></div><div>(6) 25 <= w0 + w2 + w3 + w4 + w7</div><div><br></div><div>(7) w0 + w2 + w3 + w4 + w7 <= 30</div><div><br></div><div><span style="font-family:Verdana;font-size:12px">Can I still follow your logic to convert this problem into MIPs?</span></div><div><font face="Verdana"><span style="font-size:12px">To be more precise, enumerate every value in denominators' constraints.</span></font></div><div><font face="Verdana"><span style="font-size:12px">This would result in (60 - 50 + 1) * (40 - 30 + 1) * (30 - 25 + 1) = 726 MIPs.</span></font></div><div><font face="Verdana"><span style="font-size:12px">And I choose the minimal objective value among those 726 MIPs.</span></font></div><div><font face="Verdana"><span style="font-size:12px"><br></span></font></div><div><font face="Verdana"><span style="font-size:12px">Please correct me if I'm wrong.</span></font></div><div><font face="Verdana"><span style="font-size:12px">The only thing I'm concerned about is dependencies between denominators. For example, (</span></font>w0 + w1 + w2 + w3 + w4 + w5<span style="font-size:12px;font-family:Verdana">) and (</span>w0 + w1 + w2 + w6<span style="font-family:Verdana;font-size:12px">) both contain </span><span style="font-size:12px;font-family:Verdana">(</span>w0 + w1 + w2).</div><div><br></div><div>I really appreciate your valuable help!</div><div><font face="Verdana"><span style="font-size:12px"><br></span></font></div><div><font face="Verdana"><span style="font-size:12px">Best,</span></font></div><div><font face="Verdana"><span style="font-size:12px"><br></span></font></div><div><font face="Verdana"><span style="font-size:12px">Yifan</span></font></div></div><div style="outline:none;padding:10px 0px;width:22px;margin:2px 0px 0px"><br></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Thu, Jun 18, 2020 at 11:19 PM Yifan Guan <<a href="mailto:yfguan@umich.edu" target="_blank">yfguan@umich.edu</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div dir="ltr">Hi Pietro,<div><br></div><div>First of all, thank you very much!</div><div><br></div><div>All variables are nonnegative integers. </div><div>After three days, the output of Couene is shown below:</div><div><div><img src="cid:ii_kblgmvz70" alt="Screen Shot 2020-06-18 at 8.12.41 PM.png" width="562" height="165"><br></div></div><div><br></div><div>I agree with this:</div><div>'''</div><div><span style="font-family:Verdana;font-size:12px">Probably the quickest solution is the following: given that all w variables are integer, there are only 22 values for the denominator that's in common to all expressions (unless I'm missing something from the problem). That means you can solve 22 problems (all of them MIPs, i.e. linear with integrality constraints) and choose the best solution out of these 22. Given the size of the problem, Cbc should solve all of these 22 problems quite fast (definitely less than three days).</span><br></div><div><span style="font-family:Verdana;font-size:12px">'''</span></div><div><span style="font-family:Verdana;font-size:12px"><br></span></div><div><font face="Verdana"><span style="font-size:12px">However, this is one easy special case I showed. A more general case would have different </span></font><span style="font-family:Verdana;font-size:12px">denominators.</span></div><div><span style="font-family:Verdana;font-size:12px">For example,</span></div><div><span style="font-family:Verdana;font-size:12px">Objective function: </span></div><div>abs((w0 + w1 + w2)<b><font size="4">/</font></b>(w0 + w1 + w2 + w3 + w4 + w5) - 0.7413793103448276)<span style="font-family:Verdana;font-size:12px"><br></span></div><div>+ abs((w0 + w1 + w2)<b><font size="4">/</font></b>(w0 + w1 + w2 + w3 + w4 + w5) - 0.7413793103448276)</div><div>+ abs((w0 + w1 + w2)<b><font size="4">/</font></b>(w0 + w1 + w2 + w3 + w4 + w5) - 0.7413793103448276)<br></div><div></div><div><div><br></div><div>Constraints:</div><div>(1) all variables are nonnegative integers</div><div><br></div><div>(2) 500.0 <= w0 + w1 + w2 + w3 + w4 + w5 + w6 + w7 + w8 + w9 + w10 + w11 + w12 + w13 + w14 + w15 + w16 + w17 + w18 + w19 + w20 + w21 + w22 + w23 + w24 + w25 + w26 + w27 + w28 + w29 + w30 + w31 + w32 + w33 + w34 + w35 + w36 + w37 + w38 + w39 + w40 + w41 + w42 + w43 + w44 + w45 + w46 + w47 + w48 + w49 + w50 + w51 + w52 + w53 + w54 + w55 + w56 + w57</div><div><br></div><div>(3) w0 + w1 + w2 + w3 + w4 + w5 + w6 + w7 + w8 + w9 + w10 + w11 + w12 + w13 + w14 + w15 + w16 + w17 + w18 + w19 + w20 + w21 + w22 + w23 + w24 + w25 + w26 + w27 + w28 + w29 + w30 + w31 + w32 + w33 + w34 + w35 + w36 + w37 + w38 + w39 + w40 + w41 + w42 + w43 + w44 + w45 + w46 + w47 + w48 + w49 + w50 + w51 + w52 + w53 + w54 + w55 + w56 + w57 <= 521.0</div></div><div><br></div><div><span style="font-family:Verdana;font-size:12px">Can I still follow your logic to convert this problem into MIPs?</span></div><div><font face="Verdana"><span style="font-size:12px">To be more precise,</span></font></div><div><font face="Verdana"><span style="font-size:12px"><br></span></font></div><div><font face="Verdana"><span style="font-size:12px">Please correct me if I'm wrong.</span></font></div><div><font face="Verdana"><span style="font-size:12px"><br></span></font></div><div><font face="Verdana"><span style="font-size:12px">Best,</span></font></div><div><font face="Verdana"><span style="font-size:12px"><br></span></font></div><div><font face="Verdana"><span style="font-size:12px">Yifan</span></font></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Thu, Jun 18, 2020 at 4:43 PM Pietro Belotti <<a href="mailto:pbelotti@gmx.com" target="_blank">pbelotti@gmx.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex"><div><div style="font-family:Verdana;font-size:12px"><div>Hi Yifan,</div>
<div> </div>
<div>this looks like a small problem w.r.t. variables and constraints, but the objective function is complicated. The reason why Couenne hanged is probably that after three days there was too much memory used (for instance in the branch-and-bound tree). Are all variables only integer or are they nonnegative?</div>
<div> </div>
<div>In any case, there might be some options to try but I would need the .nl file used to feed Couenne.</div>
<div> </div>
<div>What is the gap after three days? How much is the best solution and the lower bound? This might be useful.</div>
<div> </div>
<div>Probably the quickest solution is the following: given that all w variables are integer, there are only 22 values for the denominator that's in common to all expressions (unless I'm missing something from the problem). That means you can solve 22 problems (all of them MIPs, i.e. linear with integrality constraints) and choose the best solution out of these 22. Given the size of the problem, Cbc should solve all of these 22 problems quite fast (definitely less than three days).
<div> </div>
<div>Hope this helps,</div>
<div>Pietro</div>
<div>
<div name="quote" style="margin:10px 5px 5px 10px;padding:10px 0px 10px 10px;border-left:2px solid rgb(195,217,229)">
<div style="margin:0px 0px 10px"><b>Sent:</b> Monday, June 15, 2020 at 12:23 AM<br>
<b>From:</b> "Yifan Guan" <<a href="mailto:yfguan@umich.edu" target="_blank">yfguan@umich.edu</a>><br>
<b>To:</b> <a href="mailto:couenne@list.coin-or.org" target="_blank">couenne@list.coin-or.org</a><br>
<b>Subject:</b> [Couenne] Need Help: Couenne Cannot solve my MINLP problem</div>
<div name="quoted-content">
<div>Hi all,
<div> </div>
<div>Couenne cannot solve my MINLP problem.</div>
<div>Couenne ran for about three days and hanged.</div>
<div>I'm not sure why this problem occurred.</div>
<div>It might be a problem with my optimization problem formulation.</div>
<div>I hope you can give me some insightful suggestions.</div>
<div> </div>
<div>An overview of my objective function: a sum of the absolute value of probability difference. </div>
<div>In each absolute value, one probability is a numeric value. </div>
<div>And the other probability is represented by variables I want to optimize (both numerator and denominator are formed by the sum of some(or all) variables).</div>
<div> </div>
<div>Totally, 58 variables, 1 objective, and 2 constraints are declared as followed.</div>
<div> </div>
<div>My objective function:</div>
<div>minimize : </div>
<div>abs((w0 + w1 + w2 + w3 + w4 + w6 + w8 + w10 + w11 + w12 + w13 + w14 + w17 + w18 + w19 + w20 + w21 + w22 + w23 + w24 + w25 + w26 + w29 + w30 + w31 + w33 + w35 + w36 + w37 + w38 + w39 + w41 + w42 + w43 + w45 + w46 + w48 + w49 + w50 + w51 + w53 + w55 + w57)<b><font size="4">/</font></b>(w0 + w1 + w2 + w3 + w4 + w5 + w6 + w7 + w8 + w9 + w10 + w11 + w12 + w13 + w14 + w15 + w16 + w17 + w18 + w19 + w20 + w21 + w22 + w23 + w24 + w25 + w26 + w27 + w28 + w29 + w30 + w31 + w32 + w33 + w34 + w35 + w36 + w37 + w38 + w39 + w40 + w41 + w42 + w43 + w44 + w45 + w46 + w47 + w48 + w49 + w50 + w51 + w52 + w53 + w54 + w55 + w56 + w57) - 0.7413793103448276) </div>
<div>+ abs((w5 + w7 + w9 + w15 + w16 + w27 + w28 + w32 + w34 + w40 + w44 + w47 + w52 + w54 + w56)<b><font size="4">/</font></b>(w0 + w1 + w2 + w3 + w4 + w5 + w6 + w7 + w8 + w9 + w10 + w11 + w12 + w13 + w14 + w15 + w16 + w17 + w18 + w19 + w20 + w21 + w22 + w23 + w24 + w25 + w26 + w27 + w28 + w29 + w30 + w31 + w32 + w33 + w34 + w35 + w36 + w37 + w38 + w39 + w40 + w41 + w42 + w43 + w44 + w45 + w46 + w47 + w48 + w49 + w50 + w51 + w52 + w53 + w54 + w55 + w56 + w57) - 0.25862068965517243) </div>
<div>+ abs((w0 + w2 + w3 + w7 + w11 + w18 + w19 + w28 + w34 + w37 + w49 + w50 + w51 + w54 + w56 + w57)<b><font size="4">/</font></b>(w0 + w1 + w2 + w3 + w4 + w5 + w6 + w7 + w8 + w9 + w10 + w11 + w12 + w13 + w14 + w15 + w16 + w17 + w18 + w19 + w20 + w21 + w22 + w23 + w24 + w25 + w26 + w27 + w28 + w29 + w30 + w31 + w32 + w33 + w34 + w35 + w36 + w37 + w38 + w39 + w40 + w41 + w42 + w43 + w44 + w45 + w46 + w47 + w48 + w49 + w50 + w51 + w52 + w53 + w54 + w55 + w56 + w57) - 0.27586206896551724) </div>
<div>+ abs((w1 + w24 + w26 + w31 + w33 + w39 + w40 + w44 + w47)<b><font size="4">/</font></b>(w0 + w1 + w2 + w3 + w4 + w5 + w6 + w7 + w8 + w9 + w10 + w11 + w12 + w13 + w14 + w15 + w16 + w17 + w18 + w19 + w20 + w21 + w22 + w23 + w24 + w25 + w26 + w27 + w28 + w29 + w30 + w31 + w32 + w33 + w34 + w35 + w36 + w37 + w38 + w39 + w40 + w41 + w42 + w43 + w44 + w45 + w46 + w47 + w48 + w49 + w50 + w51 + w52 + w53 + w54 + w55 + w56 + w57) - 0.15517241379310345) </div>
<div>+ abs((w4 + w5 + w6 + w8 + w9 + w10 + w12 + w13 + w14 + w15 + w16 + w17 + w20 + w21 + w22 + w23 + w25 + w27 + w29 + w30 + w32 + w35 + w38 + w41 + w42 + w45 + w46 + w48 + w52 + w53 + w55)<b><font size="4">/</font></b>(w0 + w1 + w2 + w3 + w4 + w5 + w6 + w7 + w8 + w9 + w10 + w11 + w12 + w13 + w14 + w15 + w16 + w17 + w18 + w19 + w20 + w21 + w22 + w23 + w24 + w25 + w26 + w27 + w28 + w29 + w30 + w31 + w32 + w33 + w34 + w35 + w36 + w37 + w38 + w39 + w40 + w41 + w42 + w43 + w44 + w45 + w46 + w47 + w48 + w49 + w50 + w51 + w52 + w53 + w54 + w55 + w56 + w57) - 0.5344827586206896) </div>
<div>+ abs((w36 + w43)<b><font size="4">/</font></b>(w0 + w1 + w2 + w3 + w4 + w5 + w6 + w7 + w8 + w9 + w10 + w11 + w12 + w13 + w14 + w15 + w16 + w17 + w18 + w19 + w20 + w21 + w22 + w23 + w24 + w25 + w26 + w27 + w28 + w29 + w30 + w31 + w32 + w33 + w34 + w35 + w36 + w37 + w38 + w39 + w40 + w41 + w42 + w43 + w44 + w45 + w46 + w47 + w48 + w49 + w50 + w51 + w52 + w53 + w54 + w55 + w56 + w57) - 0.034482758620689655) </div>
<div>+ abs((w19 + w37 + w42)<b><font size="4">/</font></b>(w0 + w1 + w2 + w3 + w4 + w5 + w6 + w7 + w8 + w9 + w10 + w11 + w12 + w13 + w14 + w15 + w16 + w17 + w18 + w19 + w20 + w21 + w22 + w23 + w24 + w25 + w26 + w27 + w28 + w29 + w30 + w31 + w32 + w33 + w34 + w35 + w36 + w37 + w38 + w39 + w40 + w41 + w42 + w43 + w44 + w45 + w46 + w47 + w48 + w49 + w50 + w51 + w52 + w53 + w54 + w55 + w56 + w57) - 0.05172413793103448) </div>
<div>+ abs((w0 + w1 + w5 + w8 + w9 + w11 + w13 + w15 + w24 + w26 + w31 + w36 + w38 + w44 + w47 + w52 + w54)<b><font size="4">/</font></b>(w0 + w1 + w2 + w3 + w4 + w5 + w6 + w7 + w8 + w9 + w10 + w11 + w12 + w13 + w14 + w15 + w16 + w17 + w18 + w19 + w20 + w21 + w22 + w23 + w24 + w25 + w26 + w27 + w28 + w29 + w30 + w31 + w32 + w33 + w34 + w35 + w36 + w37 + w38 + w39 + w40 + w41 + w42 + w43 + w44 + w45 + w46 + w47 + w48 + w49 + w50 + w51 + w52 + w53 + w54 + w55 + w56 + w57) - 0.29310344827586204) </div>
<div>+ abs((w2 + w6 + w28)<b><font size="4">/</font></b>(w0 + w1 + w2 + w3 + w4 + w5 + w6 + w7 + w8 + w9 + w10 + w11 + w12 + w13 + w14 + w15 + w16 + w17 + w18 + w19 + w20 + w21 + w22 + w23 + w24 + w25 + w26 + w27 + w28 + w29 + w30 + w31 + w32 + w33 + w34 + w35 + w36 + w37 + w38 + w39 + w40 + w41 + w42 + w43 + w44 + w45 + w46 + w47 + w48 + w49 + w50 + w51 + w52 + w53 + w54 + w55 + w56 + w57) - 0.05172413793103448) </div>
<div>+ abs((w17 + w21 + w33 + w41 + w49 + w51)<b><font size="4">/</font></b>(w0 + w1 + w2 + w3 + w4 + w5 + w6 + w7 + w8 + w9 + w10 + w11 + w12 + w13 + w14 + w15 + w16 + w17 + w18 + w19 + w20 + w21 + w22 + w23 + w24 + w25 + w26 + w27 + w28 + w29 + w30 + w31 + w32 + w33 + w34 + w35 + w36 + w37 + w38 + w39 + w40 + w41 + w42 + w43 + w44 + w45 + w46 + w47 + w48 + w49 + w50 + w51 + w52 + w53 + w54 + w55 + w56 + w57) - 0.10344827586206896)</div>
<div>+ abs((w32 + w34 + w39 + w40)<b><font size="4">/</font></b>(w0 + w1 + w2 + w3 + w4 + w5 + w6 + w7 + w8 + w9 + w10 + w11 + w12 + w13 + w14 + w15 + w16 + w17 + w18 + w19 + w20 + w21 + w22 + w23 + w24 + w25 + w26 + w27 + w28 + w29 + w30 + w31 + w32 + w33 + w34 + w35 + w36 + w37 + w38 + w39 + w40 + w41 + w42 + w43 + w44 + w45 + w46 + w47 + w48 + w49 + w50 + w51 + w52 + w53 + w54 + w55 + w56 + w57) - 0.06896551724137931) </div>
<div>+ abs((w27 + w29 + w50)<b><font size="4">/</font></b>(w0 + w1 + w2 + w3 + w4 + w5 + w6 + w7 + w8 + w9 + w10 + w11 + w12 + w13 + w14 + w15 + w16 + w17 + w18 + w19 + w20 + w21 + w22 + w23 + w24 + w25 + w26 + w27 + w28 + w29 + w30 + w31 + w32 + w33 + w34 + w35 + w36 + w37 + w38 + w39 + w40 + w41 + w42 + w43 + w44 + w45 + w46 + w47 + w48 + w49 + w50 + w51 + w52 + w53 + w54 + w55 + w56 + w57) - 0.05172413793103448) </div>
<div>+ abs((w7 + w10 + w12 + w16 + w20 + w22 + w23 + w25 + w30 + w35 + w43 + w46 + w48 + w55 + w56)<b><font size="4">/</font></b>(w0 + w1 + w2 + w3 + w4 + w5 + w6 + w7 + w8 + w9 + w10 + w11 + w12 + w13 + w14 + w15 + w16 + w17 + w18 + w19 + w20 + w21 + w22 + w23 + w24 + w25 + w26 + w27 + w28 + w29 + w30 + w31 + w32 + w33 + w34 + w35 + w36 + w37 + w38 + w39 + w40 + w41 + w42 + w43 + w44 + w45 + w46 + w47 + w48 + w49 + w50 + w51 + w52 + w53 + w54 + w55 + w56 + w57) - 0.25862068965517243) </div>
<div>+ abs((w3 + w57)<b><font size="4">/</font></b>(w0 + w1 + w2 + w3 + w4 + w5 + w6 + w7 + w8 + w9 + w10 + w11 + w12 + w13 + w14 + w15 + w16 + w17 + w18 + w19 + w20 + w21 + w22 + w23 + w24 + w25 + w26 + w27 + w28 + w29 + w30 + w31 + w32 + w33 + w34 + w35 + w36 + w37 + w38 + w39 + w40 + w41 + w42 + w43 + w44 + w45 + w46 + w47 + w48 + w49 + w50 + w51 + w52 + w53 + w54 + w55 + w56 + w57) - 0.034482758620689655) </div>
<div>+ abs((w45 + w53)<b><font size="4">/</font></b>(w0 + w1 + w2 + w3 + w4 + w5 + w6 + w7 + w8 + w9 + w10 + w11 + w12 + w13 + w14 + w15 + w16 + w17 + w18 + w19 + w20 + w21 + w22 + w23 + w24 + w25 + w26 + w27 + w28 + w29 + w30 + w31 + w32 + w33 + w34 + w35 + w36 + w37 + w38 + w39 + w40 + w41 + w42 + w43 + w44 + w45 + w46 + w47 + w48 + w49 + w50 + w51 + w52 + w53 + w54 + w55 + w56 + w57) - 0.034482758620689655) </div>
<div>+ abs((w4 + w14 + w18)<b><font size="4">/</font></b>(w0 + w1 + w2 + w3 + w4 + w5 + w6 + w7 + w8 + w9 + w10 + w11 + w12 + w13 + w14 + w15 + w16 + w17 + w18 + w19 + w20 + w21 + w22 + w23 + w24 + w25 + w26 + w27 + w28 + w29 + w30 + w31 + w32 + w33 + w34 + w35 + w36 + w37 + w38 + w39 + w40 + w41 + w42 + w43 + w44 + w45 + w46 + w47 + w48 + w49 + w50 + w51 + w52 + w53 + w54 + w55 + w56 + w57) - 0.05172413793103448) </div>
<div>+ abs((w0 + w3 + w4 + w9 + w11 + w12 + w14 + w17 + w20 + w22 + w24 + w27 + w28 + w29 + w30 + w33 + w34 + w35 + w37 + w38 + w40 + w41 + w45 + w46 + w53 + w55)<b><font size="4">/</font></b>(w0 + w1 + w2 + w3 + w4 + w5 + w6 + w7 + w8 + w9 + w10 + w11 + w12 + w13 + w14 + w15 + w16 + w17 + w18 + w19 + w20 + w21 + w22 + w23 + w24 + w25 + w26 + w27 + w28 + w29 + w30 + w31 + w32 + w33 + w34 + w35 + w36 + w37 + w38 + w39 + w40 + w41 + w42 + w43 + w44 + w45 + w46 + w47 + w48 + w49 + w50 + w51 + w52 + w53 + w54 + w55 + w56 + w57) - 0.4482758620689655)</div>
<div>+ abs((w1 + w2 + w5 + w6 + w7 + w8 + w10 + w13 + w15 + w16 + w18 + w19 + w21 + w23 + w25 + w26 + w31 + w32 + w36 + w39 + w42 + w43 + w44 + w47 + w48 + w49 + w50 + w51 + w52 + w54 + w56 + w57)<b><font size="4">/</font></b>(w0 + w1 + w2 + w3 + w4 + w5 + w6 + w7 + w8 + w9 + w10 + w11 + w12 + w13 + w14 + w15 + w16 + w17 + w18 + w19 + w20 + w21 + w22 + w23 + w24 + w25 + w26 + w27 + w28 + w29 + w30 + w31 + w32 + w33 + w34 + w35 + w36 + w37 + w38 + w39 + w40 + w41 + w42 + w43 + w44 + w45 + w46 + w47 + w48 + w49 + w50 + w51 + w52 + w53 + w54 + w55 + w56 + w57) - 0.5517241379310345)</div>
<div> </div>
<div>Constraints:</div>
<div>(1) all variables are integer</div>
<div> </div>
<div>(2) 500.0 <= w0 + w1 + w2 + w3 + w4 + w5 + w6 + w7 + w8 + w9 + w10 + w11 + w12 + w13 + w14 + w15 + w16 + w17 + w18 + w19 + w20 + w21 + w22 + w23 + w24 + w25 + w26 + w27 + w28 + w29 + w30 + w31 + w32 + w33 + w34 + w35 + w36 + w37 + w38 + w39 + w40 + w41 + w42 + w43 + w44 + w45 + w46 + w47 + w48 + w49 + w50 + w51 + w52 + w53 + w54 + w55 + w56 + w57</div>
<div> </div>
<div>(3) w0 + w1 + w2 + w3 + w4 + w5 + w6 + w7 + w8 + w9 + w10 + w11 + w12 + w13 + w14 + w15 + w16 + w17 + w18 + w19 + w20 + w21 + w22 + w23 + w24 + w25 + w26 + w27 + w28 + w29 + w30 + w31 + w32 + w33 + w34 + w35 + w36 + w37 + w38 + w39 + w40 + w41 + w42 + w43 + w44 + w45 + w46 + w47 + w48 + w49 + w50 + w51 + w52 + w53 + w54 + w55 + w56 + w57 <= 521.0</div>
<div> </div>
<div> </div>
<div>Thank you for your time and help! I really appreciate any comments!</div>
<div> </div>
<div>Best,</div>
<div> </div>
<div>Yifan
<div> </div>
--
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<div><font color="#000000" face="arial, helvetica, sans-serif"><b>Yifan Guan</b></font>
<div style="color:rgb(136,136,136);font-size:12.8px"><font color="#000000" face="arial, helvetica, sans-serif">Computer Science Engineering</font></div>
<div style="color:rgb(136,136,136);font-size:12.8px"><span style="color:rgb(0,0,0);font-family:arial,helvetica,sans-serif;font-size:12.8px">Undergraduate University of Michigan Class of 2019</span></div>
<div style="color:rgb(136,136,136);font-size:12.8px"><span style="color:rgb(0,0,0);font-family:arial,helvetica,sans-serif;font-size:12.8px">Rackham Master </span><span style="color:rgb(0,0,0);font-family:arial,helvetica,sans-serif;font-size:12.8px">University of Michigan Class of 2021</span></div>
<div style="color:rgb(136,136,136);font-size:12.8px"><span style="color:rgb(0,0,0);font-family:arial,helvetica,sans-serif;font-size:12.8px">Website: </span><a href="https://yifanguan.github.io/" style="font-size:small" target="_blank">https://yifanguan.github.io/</a></div>
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</blockquote></div><br clear="all"><div><br></div>-- <br><div dir="ltr"><div dir="ltr"><div><div dir="ltr"><div dir="ltr"><font color="#000000" face="arial, helvetica, sans-serif"><b>Yifan Guan</b></font><br><div style="color:rgb(136,136,136);font-size:12.8px"><font face="arial, helvetica, sans-serif" color="#000000">Computer Science Engineering</font></div><div style="color:rgb(136,136,136);font-size:12.8px"><span style="color:rgb(0,0,0);font-family:arial,helvetica,sans-serif;font-size:12.8px">Undergraduate University of Michigan Class of 2019</span></div><div style="color:rgb(136,136,136);font-size:12.8px"><span style="color:rgb(0,0,0);font-family:arial,helvetica,sans-serif;font-size:12.8px">Rackham Master </span><span style="color:rgb(0,0,0);font-family:arial,helvetica,sans-serif;font-size:12.8px">University of Michigan Class of 2021</span></div><div style="color:rgb(136,136,136);font-size:12.8px"><span style="color:rgb(0,0,0);font-family:arial,helvetica,sans-serif;font-size:12.8px">Website: </span><a href="https://yifanguan.github.io/" style="font-size:small" target="_blank">https://yifanguan.github.io/</a></div></div></div></div></div></div>
</blockquote></div><br clear="all"><div><br></div>-- <br><div dir="ltr"><div dir="ltr"><div><div dir="ltr"><div dir="ltr"><font color="#000000" face="arial, helvetica, sans-serif"><b>Yifan Guan</b></font><br><div style="color:rgb(136,136,136);font-size:12.8px"><font face="arial, helvetica, sans-serif" color="#000000">Computer Science Engineering</font></div><div style="color:rgb(136,136,136);font-size:12.8px"><span style="color:rgb(0,0,0);font-family:arial,helvetica,sans-serif;font-size:12.8px">Undergraduate University of Michigan Class of 2019</span></div><div style="color:rgb(136,136,136);font-size:12.8px"><span style="color:rgb(0,0,0);font-family:arial,helvetica,sans-serif;font-size:12.8px">Rackham Master </span><span style="color:rgb(0,0,0);font-family:arial,helvetica,sans-serif;font-size:12.8px">University of Michigan Class of 2021</span></div><div style="color:rgb(136,136,136);font-size:12.8px"><span style="color:rgb(0,0,0);font-family:arial,helvetica,sans-serif;font-size:12.8px">Website: </span><a href="https://yifanguan.github.io/" style="font-size:small" target="_blank">https://yifanguan.github.io/</a></div></div></div></div></div></div></div>