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Efficient Optimization Techniques with CVX in MATLAB for Advanced Convex Programming

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This guide provides a comprehensive overview of setting up and utilizing the CVX tool for convex optimization in MATLAB. It covers crucial steps such as installation, defining variables, and formulating optimization problems between cvx_begin and cvx_end. Key topics include minimizing/maximizing convex/concave functions, working with special variables like Cvx_optval and Cvx_status, and implementing advanced techniques such as dual variables, semidefinite programming, and geometric programming. Additionally, it presents example problems related to beamforming, power allocation, and Chebyshev centers.

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Efficient Optimization Techniques with CVX in MATLAB for Advanced Convex Programming

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  1. CVX_class2014

  2. Cvx tool setup • Search for CVX tool ( http://cvxr.com/cvx/ ) • Dezip to your assigned directory • Key cvx_setup in the matlab command window No errors! cvx has been successfully installed.

  3. Cvx programming • Betweencvx_begin & cvx_end cvx_begin variables w(x,y) (complex, symmetric,…..)(refer 3.2) minimize (convex function) or Maximize (concave function) (refer 3.3) subject to …… constraints(refer to 3.4) cvx_end • Some special variables • Cvx_optval • Cvx_status • Cvx_slvtol • Cvx_slvitr

  4. Some cvx functions • Quadprog • Linprog • Norm • Norm(*,Inf) • Norm(*,1) • Refer to 3.5 and appendix B

  5. Others • Set (refer to 3.6 and appendix B.3)  • Dual variables (refer to 3.7) • Expression holders (refer to 3.8) • DCP ruleset (refer to 4) • Semidefinite programming using cvx (refer to 6) • Geometric programming using cvx(refer to 7)

  6. Q2: Chebyshev Center • Consider a polyhedron composed of the halfspaces, , , , and , please plot the maximum norm ball inside the polyhedron and show the center and the radius of it

  7. Q3:minimize the average sidelobe energy min s.t. where , .

  8. Q4: transmit beamforming (1/4)

  9. min Q4: transmit beamforming (2/4) Total power minimization Epigraph method

  10. Q4: transmit beamforming (3/4)

  11. Q4: transmit beamforming (4/4) • Angle spectrum

  12. Q5: Power allocation (a) (1/3) • Worst case design

  13. Q5: Power allocation (a) (2/3) • Epigraph form • Geometric Programming (GP) (refer to lecture 4, P4)

  14. Q5: Power allocation (a) (3/3) • Variables of change • Convex problem

  15. Q5: Power allocation (b) • Minimize the total power, subject to all the users’ SINRs are not less than • The problem can be represented as an Linear programming (LP) (refer to lecture 3, P22)

  16. Q5: Power allocation (c) • Take the worst user’s SINR in (a) in place of in (b), please re-design the transmit power , and compare with (a)

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