Nonlinear programming test cases

This is a small collection of testcases from the book In the following list there are given the data:

1. The functions defining the problem (objective function and constraints)
2. standard initial value for $x$. This latter can be changed on the input page. For some of the testcases I have changed the initial value compared to the original source in order to obtain a feasible initial $x$.
3. The optimal solution. The precision cannot be guaranteed for all given cases with the number of digits given.
4. The quotient of the largest to the smallest nonzero multiplier $u_{\max }^{*}/u_{\min }^{*}$. The problem becomes harder if this quotient is large.
5. ${\lambda }_{\max }^{*}/{\lambda }_{\min }^{*}$, the condition number of the projected Hessian, this is the condition number of the objective function seen in the tangent manifold of the feasible set at the solution. In contrast to this one the matrix
   

   ${\displaystyle {\nabla }_x^{2}L}$

   
   is not necessarily positive (semi)definite. With this quotient growing the problem becomes harder. If this value is omitted (it appears a "-") then the solution is at a vertex, which in turn makes the problem much easier. Locally it reduces to a system of $n$ nonlinear equations in $n$ unknowns for finding this vertex.

Problem 23 Number of variables: n=2 Objective function:

${\displaystyle f(x)=x_1^2+x_2^2}$

Constraints:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x_1+x_2-1}& \ge \hfill & 0\hfill \\ \multicolumn{1}{c}{x_1^2+x_2^2-1}& \ge \hfill & 0\hfill \\ \multicolumn{1}{c}{9x_1^2+x_2^2-9}& \ge \hfill & 0\hfill \\ \multicolumn{1}{c}{x_1^2-x_2}& \ge \hfill & 0\hfill \\ \multicolumn{1}{c}{x_2^2-x_1}& \ge \hfill & 0\hfill \\ \multicolumn{1}{c}{-50\le x_i}& \le \hfill & 50,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}i=1,2\hfill \end{array}}$

initial guess:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x_0}& =\hfill & (3,1)\hfill \\ \multicolumn{1}{c}{f(x_0)}& =\hfill & 10\hfill \end{array}}$

Solution:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x^{*}}& =\hfill & (1,1)\hfill \\ \multicolumn{1}{c}{f(x^{*})}& =\hfill & 2\hfill \\ \multicolumn{1}{c}{u_{\max }^{*}/u_{\min }^{*}}& =\hfill & 2/2=1\hfill \\ \multicolumn{1}{c}{{\lambda }_{\max }^{*}/{\lambda }_{\min }^{*}}& =\hfill & -\hfill \\ \multicolumn{1}{c}{u^{*}}& =\hfill & (0,0,0,2,2,0,0,0,0)\hfill \end{array}}$

Problem 33 Number of variables: n=3 Objective function:

${\displaystyle f(x)=(x_1-1)(x_1-2)(x_1-3)+x_3}$

Constraints:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x_3^2-x_2^2-x_1^2}& \ge \hfill & 0\hfill \\ \multicolumn{1}{c}{x_1^2+x_2^2+x_3^2-4}& \ge \hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \le \hfill & x_1\hfill \\ \multicolumn{1}{c}{0}& \le \hfill & x_2\hfill \\ \multicolumn{1}{c}{0}& \le \hfill & x_3\le 5\hfill \end{array}}$

initial guess:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x_0}& =\hfill & (0,0,3)\hfill \\ \multicolumn{1}{c}{f(x_0)}& =\hfill & -3\hfill \end{array}}$

Solution:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x^{*}}& =\hfill & (0,\sqrt{2},\sqrt{2})\hfill \\ \multicolumn{1}{c}{f(x^{*})}& =\hfill & \sqrt{2}-6\hfill \\ \multicolumn{1}{c}{u_{\max }^{*}/u_{\min }^{*}}& =\hfill & 11/.17678=62.23\hfill \\ \multicolumn{1}{c}{{\lambda }_{\max }^{*}/{\lambda }_{\min }^{*}}& =\hfill & -\hfill \\ \multicolumn{1}{c}{u^{*}}& =\hfill & (0.177,0.177,11,0,0,0)\hfill \end{array}}$

Problem 43 (Rosen-Suzuki) Number of variables: n=4 Objective function:

${\displaystyle f(x)=x_1^2+x_2^2+2x_3^2+x_4^2-5x_1-5x_2-21x_3+7x_4}$

Constraints:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{8-x_1^2-x_2^2-x_3^2-x_4^2-x_1+x_2-x_3+x_4}& \ge \hfill & 0\hfill \\ \multicolumn{1}{c}{10-x_1^2-2x_2^2-x_3^2-2x_4^2+x_1+x_4}& \ge \hfill & 0\hfill \\ \multicolumn{1}{c}{5-2x_1^2-x_2^2-x_3^2-2x_1+x_2+x_4}& \ge \hfill & 0\hfill \end{array}}$

initial guess:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x_0}& =\hfill & (0,0,0,0)\hfill \\ \multicolumn{1}{c}{f(x_0)}& =\hfill & 0\hfill \end{array}}$

Solution:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x^{*}}& =\hfill & (0,1,2,-1)\hfill \\ \multicolumn{1}{c}{f(x^{*})}& =\hfill & -44\hfill \\ \multicolumn{1}{c}{u_{\max }^{*}/u_{\min }^{*}}& =\hfill & 2/1=2\hfill \\ \multicolumn{1}{c}{{\lambda }_{\max }^{*}/{\lambda }_{\min }^{*}}& =\hfill & 9/8.07=1.12\hfill \\ \multicolumn{1}{c}{u^{*}}& =\hfill & (1,0,2)\hfill \end{array}}$

Problem 54 Number of variables: n=6 Objective function:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{f(x)}& =\hfill & -\exp (-h(x)/2)\hfill \\ \multicolumn{1}{c}{h(x)}& =\hfill & ((x_1-1.\text{E}4{)}^2/6.4\text{E}7+(x_1-1.\text{E}4)(x_2-1)/2.\text{E}4+6.25(x_2-1{)}^2)/.96\hfill \\ \multicolumn{1}{c}{}& \hfill & +(x_3-2.\text{E}6{)}^2/4.9\text{E}13\hfill \\ \multicolumn{1}{c}{}& \hfill & +(x_4-10{)}^2/2.5\text{E}3\hfill \\ \multicolumn{1}{c}{}& \hfill & +(x_5-1.\text{E}\text{-}3{)}^2/2.5\text{E}\text{-}3\hfill \\ \multicolumn{1}{c}{}& \hfill & +(x_6-1.\text{E}8{)}^2/2.5\text{E}17\hfill \end{array}}$

Constraints:

${\displaystyle \begin{array}{ccccccccc}\multicolumn{9}{c}{x_1+4.\text{E}3x_2-1.76\text{E}4=0}\\ \hfill 0\le & \hfill x_1\hfill & \le 2.\text{E}4\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\hfill & \hfill -10\le & \hfill x_2\hfill & \le 10\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\hfill & \hfill 0\le & \hfill x_3\hfill & \le 1.\text{E}7\hfill \\ \hfill 0\le & \hfill x_4\hfill & \le 20\hfill & \hfill -1\le & \hfill x_5\hfill & \le 1\hfill & \hfill 0\le & \hfill x_6\hfill & \le 2.\text{E}8\hfill \end{array}}$

initial guess:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x_0}& =\hfill & (6\text{E}3,1.5,4\text{E}6,2,3\text{E}\text{-}3,5\text{E}7)\hfill \\ \multicolumn{1}{c}{f(x_0)}& =\hfill & -.7651\hfill \end{array}}$

Solution:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x^{*}}& =\hfill & (91600/7,79/70,2\text{E}6,10,1\text{E}\text{-}3,1\text{E}8)\hfill \\ \multicolumn{1}{c}{f(x^{*})}& =\hfill & -\exp (-27/280)\hfill \\ \multicolumn{1}{c}{u_{\max }^{*}/u_{\min }^{*}}& =\hfill & .4865\text{E}\text{-}4/.4865\text{E}\text{-}4=1\hfill \\ \multicolumn{1}{c}{{\lambda }_{\max }^{*}/{\lambda }_{\min }^{*}}& =\hfill & 362.9/.36\text{E}\text{-}17=.10\text{E}21\hfill \\ \multicolumn{1}{c}{u^{*}}& =\hfill & (0,0,0,0,0,0,0,0,0,0,0,0,.486\text{E}\text{-}4)\hfill \end{array}}$

hs54scal is obtained from hs54 by the linear transformation

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x_1^{\text{'}}}& =\hfill & (x_1-1000)/8000\hfill \\ \multicolumn{1}{c}{x_2^{\text{'}}}& =\hfill & (x_2-1)2.5\hfill \\ \multicolumn{1}{c}{x_3^{\text{'}}}& =\hfill & (x_3-2000000)/7000000\hfill \\ \multicolumn{1}{c}{x_4^{\text{'}}}& =\hfill & (x_4-10)/50\hfill \\ \multicolumn{1}{c}{x_5^{\text{'}}}& =\hfill & (x_5-0.001)/0.05\hfill \\ \multicolumn{1}{c}{x_6^{\text{'}}}& =\hfill & (x_6-100000000)/500000000\hfill \end{array}}$

This problem is quite easy to solve. This shows the importance of proper problem scaling in practice. Problem 73 (cattle-feed) Number of variables: n=4 Objective function:

${\displaystyle f(x)=24.55x_1+26.75x_2+39x_3+40.50x_4}$

Constraints:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{2.3x_1+5.6x_2+11.1x_3+1.3x_4-5}& \ge \hfill & 0\hfill \\ \multicolumn{1}{c}{12x_1+11.9x_2+41.8x_3+52.1x_4-21}\\ \multicolumn{1}{c}{-1.645(.28x_1^2+.19x_2^2+20.5x_3^2+.62x_4^2{)}^{1/2}}& \ge \hfill & 0\hfill \\ \multicolumn{1}{c}{x_1+x_2+x_3+x_4-1}& =\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \le \hfill & x_i,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}i=1,\dots ,4\hfill \end{array}}$

initial guess:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x_0}& =\hfill & (1,1,1,1)\hfill \\ \multicolumn{1}{c}{f(x_0)}& =\hfill & 130.8\hfill \end{array}}$

Solution:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x^{*}}& =\hfill & (.6355216,-.12\text{E}\text{-}11,.3127019,.05177655)\hfill \\ \multicolumn{1}{c}{f(x^{*})}& =\hfill & 29.894378\hfill \\ \multicolumn{1}{c}{u_{\max }^{*}/u_{\min }^{*}}& =\hfill & 18.37/.2433=75.5\hfill \\ \multicolumn{1}{c}{{\lambda }_{\max }^{*}/{\lambda }_{\min }^{*}}& =\hfill & -\hfill \\ \multicolumn{1}{c}{u^{*}}& =\hfill & (.58\text{E}0,.411\text{E}0,0,.243\text{E}0,0,0,.184\text{E}2)\hfill \end{array}}$

Problem 84 Number of variables: n=5 Objective function:

${\displaystyle f(x)=-a_1-a_2{x}_1-a_3{x}_1{x}_2-a_4{x}_1{x}_3-a_5{x}_1{x}_4-a_6{x}_1{x}_5}$

Constraints:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{294000}& \ge \hfill & a_7{x}_1+a_8{x}_1{x}_2+a_9{x}_1{x}_3+a_{10}{x}_1{x}_4+a_{11}{x}_1{x}_5\ge 0\hfill \\ \multicolumn{1}{c}{294000}& \ge \hfill & a_{12}{x}_1+a_{13}{x}_1{x}_2+a_{14}{x}_1{x}_3+a_{15}{x}_1{x}_4+a_{16}{x}_1{x}_5\ge 0\hfill \\ \multicolumn{1}{c}{277200}& \ge \hfill & a_{17}{x}_1+a_{18}{x}_1{x}_2+a_{19}{x}_1{x}_3+a_{20}{x}_1{x}_4+a_{21}{x}_1{x}_5\ge 0\hfill \\ \multicolumn{1}{c}{0}& \le \hfill & x_1\le 1000\hfill \\ \multicolumn{1}{c}{1.2}& \le \hfill & x_2\le 2.4\hfill \\ \multicolumn{1}{c}{20}& \le \hfill & x_3\le 60\hfill \\ \multicolumn{1}{c}{9}& \le \hfill & x_4\le 9.3\hfill \\ \multicolumn{1}{c}{6.5}& \le \hfill & x_5\le 7\hfill \end{array}}$

The coefficients $a_i$ may be drawn from the following table:

${\displaystyle \begin{array}{cccccccc}\hfill i\hfill & \multicolumn{3}{c}{a_i}& \hfill i\hfill & \multicolumn{3}{c}{a_i}\\ \hfill 1\hfill & \hfill -24345& \hfill \hfill & \hfill & \hfill 11\hfill & \hfill 15711& \hfill .\hfill & 36\hfill \\ \hfill 2\hfill & \hfill -8720288& \hfill .\hfill & 849\hfill & \hfill 12\hfill & \hfill -155011& \hfill .\hfill & 1084\hfill \\ \hfill 3\hfill & \hfill 150512& \hfill .\hfill & 5253\hfill & \hfill 13\hfill & \hfill 4360& \hfill .\hfill & 53352\hfill \\ \hfill 4\hfill & \hfill -156& \hfill .\hfill & 6950325\hfill & \hfill 14\hfill & \hfill 12& \hfill .\hfill & 9492344\hfill \\ \hfill 5\hfill & \hfill 476470& \hfill .\hfill & 3222\hfill & \hfill 15\hfill & \hfill 10236& \hfill .\hfill & 884\hfill \\ \hfill 6\hfill & \hfill 729482& \hfill .\hfill & 8271\hfill & \hfill 16\hfill & \hfill 13176& \hfill .\hfill & 786\hfill \\ \hfill 7\hfill & \hfill -145421& \hfill .\hfill & 402\hfill & \hfill 17\hfill & \hfill -326669& \hfill .\hfill & 5104\hfill \\ \hfill 8\hfill & \hfill 2931& \hfill .\hfill & 1506\hfill & \hfill 18\hfill & \hfill 7390& \hfill .\hfill & 68412\hfill \\ \hfill 9\hfill & \hfill -40& \hfill .\hfill & 427932\hfill & \hfill 19\hfill & \hfill -27& \hfill .\hfill & 8986976\hfill \\ \hfill 10\hfill & \hfill 5106& \hfill .\hfill & 192\hfill & \hfill 20\hfill & \hfill 16643& \hfill .\hfill & 076\hfill \\ \hfill \hfill & \hfill & \hfill \hfill & \hfill & \hfill 21\hfill & \hfill 30988& \hfill .\hfill & 146\hfill \end{array}}$

initial guess:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x_0}& =\hfill & (2.52,2,37.5,9.25,6.8)\hfill \\ \multicolumn{1}{c}{f(x_0)}& =\hfill & -2351243.5\hfill \end{array}}$

Solution:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x^{*}}& =\hfill & (4.53743097,2.4,60,9.3,7)\hfill \\ \multicolumn{1}{c}{f(x^{*})}& =\hfill & -5280335.133\hfill \\ \multicolumn{1}{c}{u_{\max }^{*}/u_{\min }^{*}}& =\hfill & .7168\text{E}6/.1914\text{E}2=.37\text{E}5\hfill \\ \multicolumn{1}{c}{{\lambda }_{\max }^{*}/{\lambda }_{\min }^{*}}& =\hfill & -\hfill \\ \multicolumn{1}{c}{u^{*}}& =\hfill & (0,0,0,0,0,.191\text{E}2,0,0,0,0,0,0,\hfill \\ \multicolumn{1}{c}{}& \hfill & \hspace{0.5em}.412\text{E}5,.171\text{E}4,.717\text{E}6,.619\text{E}6)\hfill \end{array}}$

hs84scal is obtained from this by the transformation

${\displaystyle x_1^{\text{'}}=x_1/1000,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}{x}_2^{\text{'}}=x_2,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}{x}_3^{\text{'}}=x_3/10,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}{x}_4^{\text{'}}=x_4,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}{x}_5^{\text{'}}=x_5}$

and division of the first three inequalities by 1000. This means primal and dual scaling. Now the problem is much easier. Problem 86 (Colville No. 1) Number of variables: n=5 Objective function:

${\displaystyle f(x)=\sum _{j=1}^5{e}_j{x}_j+\sum _{i=1}^5\sum _{j=1}^5{c}_{\mathrm{ij}}{x}_i{x}_j+\sum _{j=1}^5{d}_j{x}_j^3}$

Constraints:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{\sum _{j=1}^5{a}_{\mathrm{ij}}{x}_j-b_i}& \ge \hfill & 0,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}i=1,\dots ,10\hfill \\ \multicolumn{1}{c}{0}& \le \hfill & x_i,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}i=1,\dots ,5\hfill \end{array}}$

The coefficients $a_{\mathrm{ij}},b_i,c_{\mathrm{ij}},d_j,e_j$ may be drawn from the following table:

${\displaystyle \begin{array}{cccccccccccccccc}j\hfill & \multicolumn{3}{c}{1}& \multicolumn{3}{c}{2}& \multicolumn{3}{c}{3}& \multicolumn{3}{c}{4}& \multicolumn{3}{c}{5}\\ e_j\hfill & \hfill -15& \hfill \hfill & \hfill & \hfill -27& \hfill \hfill & \hfill & \hfill -36& \hfill \hfill & \hfill & \hfill -18& \hfill \hfill & \hfill & \hfill -12& \hfill \hfill & \hfill \\ c_{1j}\hfill & \hfill 30& \hfill \hfill & \hfill & \hfill -20& \hfill \hfill & \hfill & \hfill -10& \hfill \hfill & \hfill & \hfill 32& \hfill \hfill & \hfill & \hfill -10& \hfill \hfill & \hfill \\ c_{2j}\hfill & \hfill -20& \hfill \hfill & \hfill & \hfill 39& \hfill \hfill & \hfill & \hfill -6& \hfill \hfill & \hfill & \hfill -31& \hfill \hfill & \hfill & \hfill 32& \hfill \hfill & \hfill \\ c_{3j}\hfill & \hfill -10& \hfill \hfill & \hfill & \hfill -6& \hfill \hfill & \hfill & \hfill 10& \hfill \hfill & \hfill & \hfill -6& \hfill \hfill & \hfill & \hfill -10& \hfill \hfill & \hfill \\ c_{4j}\hfill & \hfill 32& \hfill \hfill & \hfill & \hfill -31& \hfill \hfill & \hfill & \hfill -6& \hfill \hfill & \hfill & \hfill 39& \hfill \hfill & \hfill & \hfill -20& \hfill \hfill & \hfill \\ c_{5j}\hfill & \hfill -10& \hfill \hfill & \hfill & \hfill 32& \hfill \hfill & \hfill & \hfill -10& \hfill \hfill & \hfill & \hfill -20& \hfill \hfill & \hfill & \hfill 30& \hfill \hfill & \hfill \\ d_j\hfill & \hfill 4& \hfill \hfill & \hfill & \hfill 8& \hfill \hfill & \hfill & \hfill 10& \hfill \hfill & \hfill & \hfill 6& \hfill \hfill & \hfill & \hfill 2& \hfill \hfill & \hfill \\ a_{1j}\hfill & \hfill -16& \hfill \hfill & \hfill & \hfill 2& \hfill \hfill & \hfill & \hfill 0& \hfill \hfill & \hfill & \hfill 1& \hfill \hfill & \hfill & \hfill 0& \hfill \hfill & \hfill \\ a_{2j}\hfill & \hfill 0& \hfill \hfill & \hfill & \hfill -2& \hfill \hfill & \hfill & \hfill 0& \hfill \hfill & \hfill & \hfill & \hfill .\hfill & 4\hfill & \hfill 2& \hfill \hfill & \hfill \\ a_{3j}\hfill & \hfill -3& \hfill .\hfill & 5\hfill & \hfill 0& \hfill \hfill & \hfill & \hfill 2& \hfill \hfill & \hfill & \hfill 0& \hfill \hfill & \hfill & \hfill 0& \hfill \hfill & \hfill \\ a_{4j}\hfill & \hfill 0& \hfill \hfill & \hfill & \hfill -2& \hfill \hfill & \hfill & \hfill 0& \hfill \hfill & \hfill & \hfill -4& \hfill \hfill & \hfill & \hfill -1& \hfill \hfill & \hfill \\ a_{5j}\hfill & \hfill 0& \hfill \hfill & \hfill & \hfill -9& \hfill \hfill & \hfill & \hfill -2& \hfill \hfill & \hfill & \hfill 1& \hfill \hfill & \hfill & \hfill -2& \hfill .\hfill & 8\hfill \\ a_{6j}\hfill & \hfill 2& \hfill \hfill & \hfill & \hfill 0& \hfill \hfill & \hfill & \hfill -4& \hfill \hfill & \hfill & \hfill 0& \hfill \hfill & \hfill & \hfill 0& \hfill \hfill & \hfill \\ a_{7j}\hfill & \hfill -1& \hfill \hfill & \hfill & \hfill -1& \hfill \hfill & \hfill & \hfill -1& \hfill \hfill & \hfill & \hfill -1& \hfill \hfill & \hfill & \hfill -1& \hfill \hfill & \hfill \\ a_{8j}\hfill & \hfill -1& \hfill \hfill & \hfill & \hfill -2& \hfill \hfill & \hfill & \hfill -3& \hfill \hfill & \hfill & \hfill -2& \hfill \hfill & \hfill & \hfill -1& \hfill \hfill & \hfill \\ a_{9j}\hfill & \hfill 1& \hfill \hfill & \hfill & \hfill 2& \hfill \hfill & \hfill & \hfill 3& \hfill \hfill & \hfill & \hfill 4& \hfill \hfill & \hfill & \hfill 5& \hfill \hfill & \hfill \\ a_{10j}\hfill & \hfill 1& \hfill \hfill & \hfill & \hfill 1& \hfill \hfill & \hfill & \hfill 1& \hfill \hfill & \hfill & \hfill 1& \hfill \hfill & \hfill & \hfill 1& \hfill \hfill & \hfill \\ b_j\hfill & \hfill -40& \hfill \hfill & \hfill & \hfill -2& \hfill \hfill & \hfill & \hfill -& \hfill .\hfill & 25\hfill & \hfill -4& \hfill \hfill & \hfill & \hfill -4& \hfill \hfill & \hfill \\ b_{5+j}\hfill & \hfill -1& \hfill \hfill & \hfill & \hfill -40& \hfill \hfill & \hfill & \hfill -60& \hfill \hfill & \hfill & \hfill 5& \hfill \hfill & \hfill & \hfill 1& \hfill \hfill & \hfill \end{array}}$

initial guess:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x_0}& =\hfill & (0,0,0,0,1)\hfill \\ \multicolumn{1}{c}{f(x_0)}& =\hfill & 20\hfill \end{array}}$

Solution:

${\displaystyle \begin{array}{ccc}\hfill x^{*}& \hfill =\hfill & (.3,.33346761,.4,.42831010,.22396487)\hfill \\ \hfill f(x^{*})& \hfill =\hfill & -32.34867897\hfill \\ \hfill u_{\max }^{*}/u_{\min }^{*}& \hfill =\hfill & 11.84/.1039=113.9\hfill \\ \hfill {\lambda }_{\max }^{*}/{\lambda }_{\min }^{*}& \hfill =\hfill & 68.1/68.1=1\hfill \\ \hfill u^{*}& \hfill =\hfill & (\begin{array}{ccccc}.0\hfill & ,\hspace{0.5em}.0\hfill & ,\hspace{0.5em}.517\text{E}1\hfill & ,\hspace{0.5em}.0\hfill & ,\hfill \\ .306\text{E}1\hfill & ,\hspace{0.5em}.118\text{E}2\hfill & ,\hspace{0.5em}.0\hfill & ,\hspace{0.5em}.0\hfill & ,\hfill \\ .104\text{E}0\hfill & ,\hspace{0.5em}.0\hfill & ,\hspace{0.5em}.0\hfill & ,\hspace{0.5em}.0\hfill & ,\hfill \\ .0\hfill & ,\hspace{0.5em}.0\hfill & ,\hspace{0.5em}.0\hspace{0.5em})\hfill \end{array}\hfill \end{array}}$

Problem 93 (transformer design) Number of variables: n=6 Objective function:

${\displaystyle \begin{array}{ccc}\hfill f(x)& \hfill =\hfill & .0204x_1{x}_4(x_1+x_2+x_3)+.0187x_2{x}_3(x_1+1.57x_2+x_4)\hfill \\ \hfill & \hfill \hfill & +.0607x_1{x}_4{x}_5^2(x_1+x_2+x_3)+.0437x_2{x}_3{x}_6^2(x_1+1.57x_2+x_4)\hfill \end{array}}$

Constraints:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{.001x_1{x}_2{x}_3{x}_4{x}_5{x}_6-2.07}& \ge \hfill & 0\hfill \\ \multicolumn{1}{c}{1-.00062x_1{x}_4{x}_5^2(x_1+x_2+x_3)-.00058x_2{x}_3{x}_6^2(x_1+1.57x_2+x_4)}& \ge \hfill & 0\hfill \\ \multicolumn{1}{c}{0\le x_i\hspace{0.5em},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}i=1,\dots ,6}\end{array}}$

initial guess:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x_0}& =\hfill & (5.54,4.4,12.02,11.82,.702,.852)\hfill \\ \multicolumn{1}{c}{f(x_0)}& =\hfill & 137.066\hfill \end{array}}$

Solution:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x^{*}}& =\hfill & (5.332666,4.656744,10.43299,12.08230,.7526074,.87865084)\hfill \\ \multicolumn{1}{c}{f(x^{*})}& =\hfill & 135.075961\hfill \\ \multicolumn{1}{c}{u_{\max }^{*}/u_{\min }^{*}}& =\hfill & 71.46/62.15=1.15\hfill \\ \multicolumn{1}{c}{{\lambda }_{\max }^{*}/{\lambda }_{\min }^{*}}& =\hfill & 118.9/.21=562.9\hfill \\ \multicolumn{1}{c}{u^{*}}& =\hfill & (.715\text{E}2,.622\text{E}2,.0,.0,.0,.0,.0,.0)\hfill \end{array}}$

Problem 98 Number of variables: n=6 Objective function:

${\displaystyle f(x)=4.3x_1+31.8x_2+63.3x_3+15.8x_4+68.5x_5+4.7x_6}$

Constraints:

${\displaystyle \begin{array}{c}\multicolumn{1}{c}{17.1x_1+38.2x_2+204.2x_3+212.3x_4+623.4x_5+1495.5x_6}\\ \multicolumn{1}{c}{-169x_1{x}_3-3580x_3{x}_5-3810x_4{x}_5-18500x_4{x}_6-24300x_5{x}_6}& \ge \hfill & b_1\hfill \\ \multicolumn{1}{c}{17.9x_1+36.8x_2+113.9x_3+169.7x_4+337.8x_5+1385.2x_6}\\ \multicolumn{1}{c}{-139x_1{x}_3-2450x_4{x}_5-16600x_4{x}_6-17200x_5{x}_6}& \ge \hfill & b_2\hfill \\ \multicolumn{1}{c}{-273x_2-70x_4-819x_5+26000x_4{x}_5}& \ge \hfill & b_3\hfill \\ \multicolumn{1}{c}{159.9x_1-311x_2+587x_4+391x_5+2198x_6-14000x_1{x}_6}& \ge \hfill & b_4\hfill \end{array}}$

${\displaystyle \begin{array}{ccc}\hfill b_1& \hfill =\hfill & 32.97\hfill \\ \hfill b_2& \hfill =\hfill & 25.12\hfill \\ \hfill b_3& \hfill =\hfill & -124.08\hfill \\ \hfill b_4& \hfill =\hfill & -173.02\hfill \end{array}}$

${\displaystyle \begin{array}{ccc}0\le x_1\le .31\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\hfill & ,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}0\le x_3\le .068\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\hfill & ,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}0\le x_5\le .028\hfill \\ 0\le x_2\le .046\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\hfill & ,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}0\le x_4\le .042\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\hfill & ,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}0\le x_6\le .0134\hfill \end{array}}$

initial guess:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x_0}& =\hfill & (0,0,0,0,0,0)\hfill \\ \multicolumn{1}{c}{f(x_0)}& =\hfill & 0\hfill \end{array}}$

Solution:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x^{*}}& =\hfill & (.2685649,0,0,0,.028,.0134)\hfill \\ \multicolumn{1}{c}{f(x^{*})}& =\hfill & 3.1358091\hfill \\ \multicolumn{1}{c}{u_{\max }^{*}/u_{\min }^{*}}& =\hfill & 200/.251=.8\text{E}3\hfill \\ \multicolumn{1}{c}{{\lambda }_{\max }^{*}/{\lambda }_{\min }^{*}}& =\hfill & -\hfill \end{array}}$

${\displaystyle u^{*}=(\begin{array}{cccccccc}.251\text{E}0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill \\ 0.\hfill & \hfill ,\hfill & .222\text{E}2\hfill & \hfill ,\hfill & .486\text{E}2\hfill & \hfill ,\hfill & .516\text{E}2\hfill & \hfill ,\hfill \\ 0.\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill \\ .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .638\text{E}1\hfill & \hfill ,\hfill & .200\text{E}3\hfill & \hfill )\hfill \end{array}}$

Problem 99 Number of variables: n=7 Objective function:

${\displaystyle f(x)=r_8(x{)}^2}$

${\displaystyle r_1(x)=0,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{r}_i(x)=a_i(t_i-t_{i-1})\cos x_{i-1}+r_{i-1}(x),\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}i=2,\dots ,8}$

Constraints:

${\displaystyle \begin{array}{ccc}\hfill 0\le x_i\le 1.58,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}& \hfill \hfill & i=1,\dots ,7\hfill \\ \hfill q_8(x)-1.\text{E}5& \hfill =\hfill & 0\hfill \\ \hfill s_8(x)-1.\text{E}3& \hfill =\hfill & 0\hfill \\ \hfill q_1(x)=s_1(x)& \hfill =\hfill & 0\hfill \\ \hfill q_i(x)& \hfill =\hfill & .5(t_i-t_{i-1}{)}^2(a_i\sin x_{i-1}-b)+(t_i-t_{i-1})s_{i-1}(x)+q_{i-1}(x)\hfill \\ \hfill s_i(x)& \hfill =\hfill & (t_i-t_{i-1})(a_i\sin x_{i-1}-b)+s_{i-1}(x),\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}i=2,\dots ,8\hfill \end{array}}$

The coefficients $a_i$ and $t_i$ are

${\displaystyle \begin{array}{ccc}\hfill i& \hfill a_i& \hfill t_i\\ \hfill 1& \hfill 0& \hfill 0\\ \hfill 2& \hfill 50& \hfill 25\\ \hfill 3& \hfill 50& \hfill 50\\ \hfill 4& \hfill 75& \hfill 100\\ \hfill 5& \hfill 75& \hfill 150\\ \hfill 6& \hfill 75& \hfill 200\\ \hfill 7& \hfill 100& \hfill 290\\ \hfill 8& \hfill 100& \hfill 380\\ \multicolumn{3}{c}{b=32}\end{array}}$

initial guess:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x_0}& =\hfill & (.5,.5,.5,.5,.5,.5,.5)\hfill \\ \multicolumn{1}{c}{f(x_0)}& =\hfill & -.7763605\text{E}9\hfill \end{array}}$

Solution:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x^{*}}& =\hfill & (.5424603,.5290159,.5084506,.4802693,.4512352,.4091878,.3527847)\hfill \\ \multicolumn{1}{c}{f(x^{*})}& =\hfill & -.831079892\text{E}9\hfill \\ \multicolumn{1}{c}{u_{\max }^{*}/u_{\min }^{*}}& =\hfill & .1934\text{E}5/.4194\text{E}2=.46\text{E}3\hfill \\ \multicolumn{1}{c}{{\lambda }_{\max }^{*}/{\lambda }_{\min }^{*}}& =\hfill & .50\text{E}9/.84\text{E}8=5.95\hfill \end{array}}$

${\displaystyle \begin{array}{ccc}\hfill u^{*}& \hfill =\hfill & (\begin{array}{cccccccc}.0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill \\ .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill \\ .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill \\ .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .419\text{E}2\hfill & \hfill ,\hfill & .193\text{E}5\hfill & \hfill )\hfill \end{array}\hfill \end{array}}$

hs99scal is obtained from this one by the transformation (dual scaling)

${\displaystyle f\to f/{10}^9,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{h}_1\to h_1/{10}^5,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{h}_2\to h_2/{10}^3\hspace{0.5em}.}$

Problem 112 (chemical equilibrium) Number of variables: n=10 Objective function:

${\displaystyle f(x)=\sum _{j=1}^{10}{x}_j(c_j+\ln \frac{x_j}{x_1+\dots +x_{10}})}$

${\displaystyle \begin{array}{cccc}\hfill j\hfill & \hfill c_j& \hfill j& \hfill c_j\\ \hfill 1\hfill & \hfill -6.089& \hfill 6& \hfill -14.986\\ \hfill 2\hfill & \hfill -17.164& \hfill 7& \hfill -24.100\\ \hfill 3\hfill & \hfill -34.054& \hfill 8& \hfill -10.708\\ \hfill 4\hfill & \hfill -5.914& \hfill 9& \hfill -26.662\\ \hfill 5\hfill & \hfill -24.721& \hfill \mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}10& \hfill -22.179\end{array}}$

Constraints:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x_1+2x_2+2x_3+x_6+x_{10}-2}& =\hfill & 0\hfill \\ \multicolumn{1}{c}{x_4+2x_5+x_6+x_7-1}& =\hfill & 0\hfill \\ \multicolumn{1}{c}{x_3+x_7+x_8+2x_9+x_{10}-1}& =\hfill & 0\hfill \\ \multicolumn{1}{c}{1.\text{E}\text{-}6\le x_i,}& \hfill & i=1,\dots ,10\hfill \end{array}}$

initial guess:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x_0}& =\hfill & (.1,\dots ,.1)\hfill \\ \multicolumn{1}{c}{f(x_0)}& =\hfill & -20.961\hfill \end{array}}$

Solution:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x^{*}}& =\hfill & (.01773548,.08200180,.8825646,.723325\text{E}\text{-}3,\hfill \\ \multicolumn{1}{c}{}& \hfill & \mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}.4907851,.4335469\text{E}\text{-}3,.01727298,\hfill \\ \multicolumn{1}{c}{}& \hfill & \mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}.007765639,.01984929,.05269826)\hfill \\ \multicolumn{1}{c}{f(x^{*})}& =\hfill & -47.707579\hfill \\ \multicolumn{1}{c}{u_{\max }^{*}/u_{\min }^{*}}& =\hfill & 15.02/.262\text{E}\text{-}3=.57\text{E}5\hfill \\ \multicolumn{1}{c}{{\lambda }_{\max }^{*}/{\lambda }_{\min }^{*}}& =\hfill & 191/8.98=21.3\hfill \\ \multicolumn{1}{c}{}\end{array}}$

${\displaystyle u^{*}=(\begin{array}{cccccccc}.0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .262\text{E}\text{-}3\hfill & \hfill ,\hfill \\ .0\hfill & \hfill ,\hfill & .130\text{E}\text{-}2\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill \\ .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & -.958\text{E}1\hfill & \hfill ,\hfill & -.126\text{E}2\hfill & \hfill ,\hfill \\ -1.150\text{E}2)\hfill \end{array}}$

Problem 114 (alkylation process) Number of variables: n=10 Objective function:

${\displaystyle f(x)=5.04x_1+.035x_2+10x_3+3.36x_5-.063x_4{x}_7}$

Constraints:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{g_1(x)}& =\hfill & 35.82-.222x_{10}-{\mathrm{bx}}_9\ge 0\hfill \\ \multicolumn{1}{c}{g_2(x)}& =\hfill & -133+3x_7-{\mathrm{ax}}_{10}\ge 0\hfill \\ \multicolumn{1}{c}{g_3(x)}& =\hfill & -g_1(x)+x_9(1/b-b)\ge 0\hfill \\ \multicolumn{1}{c}{g_4(x)}& =\hfill & -g_2(x)+(1/a-a)x_{10}\ge 0\hfill \\ \multicolumn{1}{c}{g_5(x)}& =\hfill & 1.12x_1+.13167x_1{x}_8-.00667x_1{x}_8^2-{\mathrm{ax}}_4\ge 0\hfill \\ \multicolumn{1}{c}{g_6(x)}& =\hfill & 57.425+1.098x_8-.038x_8^2+.325x_6-{\mathrm{ax}}_7\ge 0\hfill \\ \multicolumn{1}{c}{g_7(x)}& =\hfill & -g_5(x)+(1/a-a)x_4\ge 0\hfill \\ \multicolumn{1}{c}{g_8(x)}& =\hfill & -g_6(x)+(1/a-a)x_7\ge 0\hfill \\ \multicolumn{1}{c}{h_1(x)}& =\hfill & 1.22x_4-x_1-x_5=0\hfill \\ \multicolumn{1}{c}{h_2(x)}& =\hfill & 98000x_3/(x_4{x}_9+1000x_3)-x_6=0\hfill \\ \multicolumn{1}{c}{h_3(x)}& =\hfill & (x_2+x_5)/x_1-x_8=0\hfill \\ \multicolumn{1}{c}{a}& =\hfill & 0.99\hfill \\ \multicolumn{1}{c}{b}& =\hfill & .9\hfill \end{array}}$

${\displaystyle \begin{array}{ccccc}\hfill .00001& \hfill \le \hfill & x_1\hfill & \hfill \le \hfill & 2000\hfill \\ \hfill .00001& \hfill \le \hfill & x_2\hfill & \hfill \le \hfill & 16000\hfill \\ \hfill .00001& \hfill \le \hfill & x_3\hfill & \hfill \le \hfill & 120\hfill \\ \hfill .00001& \hfill \le \hfill & x_4\hfill & \hfill \le \hfill & 5000\hfill \\ \hfill .00001& \hfill \le \hfill & x_5\hfill & \hfill \le \hfill & 2000\hfill \\ \hfill 85& \hfill \le \hfill & x_6\hfill & \hfill \le \hfill & 93\hfill \\ \hfill 90& \hfill \le \hfill & x_7\hfill & \hfill \le \hfill & 95\hfill \\ \hfill 3& \hfill \le \hfill & x_8\hfill & \hfill \le \hfill & 12\hfill \\ \hfill 1.2& \hfill \le \hfill & x_9\hfill & \hfill \le \hfill & 4\hfill \\ \hfill 145& \hfill \le \hfill & x_{10}\hfill & \hfill \le \hfill & 162\hfill \end{array}}$

initial guess:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x_0}& =\hfill & (1745,12000,110,3048,1974,89.2,92.8,8,3.6,145)\hfill \\ \multicolumn{1}{c}{f(x_0)}& =\hfill & -872.3872\hfill \\ \multicolumn{1}{c}{}\end{array}}$

Solution:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x^{*}}& =\hfill & (1698.096,15818.73,54.10228,3031.226,2000,\hfill \\ \multicolumn{1}{c}{}& \hfill & \hspace{0.5em}90.11537,95,10.49336,1.561636,153.53535)\hfill \\ \multicolumn{1}{c}{f(x^{*})}& =\hfill & -1768.80696\hfill \\ \multicolumn{1}{c}{u_{\max }^{*}/u_{\min }^{*}}& =\hfill & 311.8/.6778=460\hfill \\ \multicolumn{1}{c}{{\lambda }_{\max }^{*}/{\lambda }_{\min }^{*}}& =\hfill & .14\text{E}\text{-}4/.14\text{E}\text{-}4=1\hfill \\ \multicolumn{1}{c}{}\end{array}}$

${\displaystyle u^{*}=(\begin{array}{cccccccc}.0\hfill & \hfill ,\hfill & .699\text{E}2\hfill & \hfill ,\hfill & .312\text{E}3\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill \\ .678\text{E}0\hfill & \hfill ,\hfill & .230\text{E}3\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill \\ .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill \\ .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill \\ .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill \\ .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .884\text{E}0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill \\ .173\text{E}3\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill \\ -.421\text{E}1\hfill & \hfill ,\hfill & .746\text{E}2\hfill & \hfill ,\hfill & .594\text{E}2\hfill & \hfill )\hfill \end{array}}$

Problem 118 (a QP) Number of variables: n=15 Objective function:

${\displaystyle f(x)=\sum _{k=0}^4(2.3x_{3k+1}+.0001x_{3k+1}^2+1.7x_{3k+2}+.0001x_{3k+2}^2+2.2x_{3k+3}+.00015x_{3k+3}^2)}$

Constraints:

${\displaystyle \begin{array}{cccc}\hfill 0\le x_{3j+1}-x_{3j-2}+7& \hfill \le \hfill & 13\hfill & \mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}0\le x_{3j+2}-x_{3j-1}+7\le 14\hfill \\ \hfill 0\le x_{3j+3}-x_{3j}+7& \hfill \le \hfill & 13\hfill & \mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}j=1,\dots ,4\hfill \\ \hfill x_1+x_2+x_3-60& \hfill \ge \hfill & 0\hfill \\ \hfill x_4+x_5+x_6-50& \hfill \ge \hfill & 0\hfill \\ \hfill x_7+x_8+x_9-70& \hfill \ge \hfill & 0\hfill \\ \hfill x_{10}+x_{11}+x_{12}-85& \hfill \ge \hfill & 0\hfill \\ \hfill x_{13}+x_{14}+x_{15}-100& \hfill \ge \hfill & 0\hfill \\ \multicolumn{4}{c}{\begin{array}{ccc}8\le x_1\le 21\hfill & 43\le x_2\le 57\hfill & 3\le x_3\le 16\hfill \\ 0\le x_{3k+1}\le 90\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\hfill & 0\le x_{3k+2}\le 120\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}\hfill & 0\le x_{3k+3}\le 60\hfill \\ k=1,\dots ,4\hfill \end{array}}\end{array}}$

initial guess:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x_0}& =\hfill & (20,55,15,20,60,20,20,60,20,20,60,20,20,60,20)\hfill \\ \multicolumn{1}{c}{f(x_0)}& =\hfill & 942.716\hfill \end{array}}$

Solution:

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{x^{*}}& =\hfill & (8,49,3,1,56,0,1,63,6,3,70,12,5,77,18)\hfill \\ \multicolumn{1}{c}{f(x^{*})}& =\hfill & 664.8204500\hfill \\ \multicolumn{1}{c}{u_{\max }^{*}/u_{\min }^{*}}& =\hfill & 2.941/.04860=60.5\hfill \\ \multicolumn{1}{c}{{\lambda }_{\max }^{*}/{\lambda }_{\min }^{*}}& =\hfill & -\hfill \\ \multicolumn{1}{c}{}\end{array}}$

${\displaystyle u^{*}=(\begin{array}{cccccccc}.230\text{E}1\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill \\ .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill \\ .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill \\ .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill \\ .486\text{E}\text{-}1\hfill & \hfill ,\hfill & .176\text{E}1\hfill & \hfill ,\hfill & .0117\text{E}1\hfill & \hfill ,\hfill & .586\text{E}0\hfill & \hfill ,\hfill \\ .0\hfill & \hfill ,\hfill & .291\text{E}0\hfill & \hfill ,\hfill & .193\text{E}0\hfill & \hfill ,\hfill & .956\text{E}\text{-}1\hfill & \hfill ,\hfill \\ .166\text{E}1\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .230\text{E}1\hfill & \hfill ,\hfill & .230\text{E}1\hfill & \hfill ,\hfill \\ .230\text{E}1\hfill & \hfill ,\hfill & .294\text{E}1\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .540\text{E}0\hfill & \hfill ,\hfill \\ .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .191\text{E}1\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill \\ .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill \\ .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill \\ .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill \\ .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill \\ .0\hfill & \hfill ,\hfill & .0\hfill & \hfill ,\hfill & .0\hfill & \hfill )\hfill & \hfill \end{array}}$

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On 16 Jun 2016, 16:46.