Many nonconvex quadratic programming benchmark problems are provided in [22]. These problems cannot be solved with the q

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Many Nonconvex Quadratic Programming Benchmark Problems Are Provided In 22 These Problems Cannot Be Solved With The Q 1 (39.08 KiB) Viewed 39 times

Many Nonconvex Quadratic Programming Benchmark Problems Are Provided In 22 These Problems Cannot Be Solved With The Q 2 (45.26 KiB) Viewed 39 times

Many nonconvex quadratic programming benchmark problems are provided in [22]. These problems cannot be solved with the quadprog() function discussed in Chap- ter 4. Therefore, trials on these problems can be made with the general-purpose non- linear programming problem solvers. In this section, examples are given to show the global optimum solutions. Example 5.13. Solve the nonconvex quadratic programming problem studied in Ex- ample 4.30. For the convenience of presentation, the mathematical model is given again: c²x+d¹y-¹x¹Qx, min x s.t. 2x₂+2x₂+Y+Y₂10 2X₁+2x3+Y6+YB10 2x₂+2x+y+YB<10 -8X₁+Y60 -8X₂+Y70 -8X3+Yg <0 -2X4-Y1+Y60 -2₂-₁+Y₁0 -2Y4-Y5+YBO 0X, 1, 1=1,2,3,4 |0y,1, 1=1,2,3,4,5,9 [₁20, 1=6,7,8 where c = [5,5,5, 5], d = [-1,-1,-1, −1, −1, −1, −1, −1,-1], Q = 101.
A powerful solver for nonlinear programming problems is provided in MATLAB Opti- mization Toolbox. The seemingly complicated nonlinear programming problems can be solved easily with the provided tools, since the standard form of nonlinear pro- gramming problems can be entered into MATLAB environment, so as to find the solu- tions directly. The biggest problem of the solver is that sometimes it may depend too much upon the use of initial search point. If the initial point is not selected properly, the global optimum solutions cannot be found. There are to date no well-accepted general-purpose methods in initial point selection, so trials must be made when find- ing global optimum solutions. An attempt is made in this section. Based on the idea, a general-purpose global nonlinear programming problem solver is provided. Besides, further attempts are made to two classes of problems - global optimum solutions of nonconvex quadratic programming and concave-cost transportation problems.