A library of examples for CVX

The following is a catalog of usage examples for CVX, a Matlab-based modeling system for disciplined convex programming.

Each page includes the full code for a single example, followed by a listing of the output generated by MATLAB when the code is run. For each page we also provide a link to a downloadable copy of the code which you can run and modify yourself. Of course, to do so you will need to have CVX installed; so if necessary please proceed to the main CVX page to download the distribution.

These examples are public domain. You are free to use them in any way you wish; but when you do, we request that you give appropriate credit to the authors. A number of people have contributed to the examples in this library, including Lieven Vandenberghe, Joëlle Skaf, Argyris Zymnis, Almir Mutapcic, Michael Grant, and Stephen Boyd.

If your browser supports JavaScript, then the list below will be dynamic. Clicking on the icons will expand/collapse directories to reveal/hide their contents.

Last updated: 26-Oct-2006

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• Antenna array design

  Relevant links: ~boyd/ant-pat.html ~boyd/magdes.html

  • Minimize beamwidth of an array with arbitrary 2-D geometry (ant_array_min_beamwidth.m)
  • Minimize sidelobe level of a uniform linear array via spectral factorization (line_array_spec_fact.m)
  • Minimize sidelobe level of an FIR broadband far-field antenna array (broadband_array_min_sidelobe.m)
  • Minimize sidelobe level of an array with arbitrary 2-D geometry (ant_array_min_sidelobe.m)
  • Minimize thermal noise power of an array with arbitrary 2-D geometry (ant_array_min_therm_noise.m)
  • Utility functions:
    • Plot a polar plot of an antenna array sensitivity (polar_plot_ant.m)
    • Spectral factorization using Kolmogorov 1939 approach (spectral_fact.m)
• Circuit design

  Relevant links: ~boyd/rc.html ~boyd/rc_iccad.html ~boyd/date05.html ~boyd/gp_digital_ckt.html

  • Combined sizing of drivers, repeaters, and wire (wire_driver_sizing.m)
  • Combined wire sizing and spacing (wire_sizing_spacing.m)
  • Digital circuit sizing example (GP) (dig_ckt_sizing.m)
  • Digital circuit sizing for an inverter chain (GP) (inverter_chain_sizing.m)
  • Elmore delay sizing for a straight wire (GP) (elmore_straight_wire.m)
  • LC oscillator design (GP) (LC_osc_design.m)
  • Optimal interconnect wire sizing (wire_sizing.m)
  • Sizing of clock meshes (clock_mesh.m)
  • Tri-state bus sizing and topology design (tristate_bus_sizing.m)
  • Two-input NAND gate sizing (GP) (simple_NAND2_gate_design.m)
  • Wire sizing and topology design (wire_sizing_topology.m)
  • Utility functions:
    • Computes the step response of a linear system (simple_step.m)
    • Plots four different taper desings on a single graph. (plot_four_tapers.m)
• Figures, examples, and exercises from Boyd & Vandenberghe, "Convex Optimization"

  Visit the book's web site for more info: ~boyd/cvxbook

  • Chapter 4: Convex optimization problems
    • Section 4.3.1: Compute and display the Chebyshev center of a 2D polyhedron (chebyshev_center_2D.m)
    • Section 4.3.1: Compute the Chebyshev center of a polyhedron (chebyshev_center.m)
    • Section 4.5.4: Design of a cantilever beam: recursive formulation (GP) (cantilever_beam_rec.m)
    • Section 4.5.4: Frobenius norm diagonal scaling (GP) (frob_norm_diag_scaling.m)
    • Section 4.5.4: Minimum spectral radius via Peron-Frobenius theory (GP) (min_spec_rad_ppl_dynamics.m)
    • Section 4.6.3: Find the fastest mixing Markov chain on a graph (fastest_mixing_MC.m)
    • Exercise 4.27: Matrix fractional minimization using second-order cone programming (ex_4_27.m)
    • Exercise 4.31: Design of a cantilever beam (GP) (cantilever_beam.m)
    • Exercise 4.38(b): Linear matrix inequalities with one variable (ex_4_38.m)
    • Exercise 4.3: Solve a simple QP with inequality constraints (ex_4_3.m)
    • Exercise 4.47: Maximum determinant PSD matrix completion (max_det_psd_completion.m)
    • Exercise 4.5: Show the equivalence of 3 convex problem formations (ex_4_5.m)
    • Utility functions:
      • Plots a cantilever beam as a 3D figure. (cantilever_beam_plot.m)
  • Chapter 5: Duality
    • Section 5.2.4: Solves a simple QCQP (qcqp.m)
    • Section 5.2.5: Mixed strategies for matrix games (matrix_games.m)
    • Section 5.2.5: Mixed strategies for matrix games (LP formulation) (matrix_games_LP.m)
    • Examples 5.6,5.8: An l_p norm approximation problem (norm_approx.m)
    • Exercise 5.19c: Markovitz portfolio optimization w/ diversification constraint (ex_5_19.m)
    • Exercise 5.1d: Sensitivity analysis for a simple QCQP (ex_5_1.m)
    • Exercise 5.33: Parametrized l1-norm approximation (ex_5_33.m)
    • Exercise 5.39: SDP relaxations of the two-way partitioning problem (ex_5_39.m)
  • Chapter 6: Approximation and fitting
    • Section 6.1.2: Residual minimization with deadzone penalty (deadzone.m)
    • Example 6.2: Robust regression using the Huber penalty (fig6_5.m)
    • Example 6.3: Optimal input design (fig6_6.m)
    • Example 6.4: Regressor selection problem (regressor_cvx.m)
    • Example 6.6: Comparison of worst-case robust, Tikhonov, and nominal least squares (wcrobls.m)
    • Example 6.8: Spline fitting (fig6_20.m)
    • Example 6.9: Bounding consumer preference (preference_regions.m)
    • Figure 6.15: A comparison of stochastic and worst-case robust approximation (fig6_15.m)
    • Figure 6.19: Polynomial fitting (fig6_19.m)
    • Figure 6.24: Fitting a convex function to given data (convex_interpolation.m)
    • Figure 6.2: Penalty function approximation (penalty_comp_cvx.m)
    • Figure 6.9: An optimal tradeoff curve (fig6_9.m)
    • Figures 6.11-6.14: Total variation reconstruction (tv_cvx.m)
    • Figures 6.21-6.23: Basis pursuit using Gabor functions (basispursuit.m)
    • Figures 6.8-6.10: Quadratic smoothing (smoothrec_cvx.m)
  • Chapter 7: Statistical estimation
    • Section 7.4.3: Probability bounds example with Voronoi diagram (probbounds.m)
    • Section 7.5.2: Experiment design (expdesign.m)
    • Example 7.2: Maximum entropy distribution (maxent.m)
    • Example 7.4: Binary hypothesis testing (detector2.m)
    • Figure 7.1: Logistic regression (logistics.m)
    • Figure 7.1: Logistic regression (GP version) (logistics_gp.m)
    • Utility functions:
      • Computes Chebyshev lower bounds on probability vectors (cheb.m)
      • Computes Chernoff upper bounds on probability (cher.m)
      • estimates probability than random vector x in R2 (montecarlo.m)
  • Chapter 8: Geometric problems
    • Section 8.1.1: Separating a point from a polyhedron (separate_pt_poly.m)
    • Section 8.2.2: Separating polyhedra in 2D (separate_poly_2D.m)
    • Example 8.3: Bounding correlation coefficients (ex_8_3.m)
    • Example 8.4: One free point localization (ex_8_4.m)
    • Example 8.7: Floorplan generation test script (test_floorplan.m)
    • Figure 8.10: Approximate linear discrimination via linear programming (svm_1.m)
    • Figure 8.11: Approximate linear discrimination via support vector classifier (svm_2.m)
    • Figure 8.15: Linear placement problem (placement_lin.m)
    • Figure 8.16: Quadratic placement problem (placement_quad.m)
    • Figure 8.17: Fourth-order placement problem (placement_quar.m)
    • Figure 8.8: Simplest linear discrimination (linear_discr.m)
    • Figure 8.9: Robust linear discrimination problem (robust_lin_discr.m)
    • Euclidean distance between polyhedra (eucl_dist_poly.m)
    • Euclidean distance between polyhedra in 2D (eucl_dist_poly_2D.m)
    • Euclidean projection on a halfspace (eucl_proj_hlf.m)
    • Euclidean projection on a hyperplane (eucl_proj_hyp.m)
    • Euclidean projection on a rectangle (eucl_proj_rect.m)
    • Euclidean projection on the nonnegative orthant (eucl_proj_cone1.m)
    • Euclidean projection on the semidefinite cone (eucl_proj_cone2.m)
    • Floor planning (floor_plan.m)
    • Maximum volume inscribed ellipsoid in a polyhedron (max_vol_ellip_in_polyhedra.m)
    • Minimum volume ellipsoid covering a finite set (min_vol_elp_finite_set.m)
    • Minimum volume ellipsoid covering union of ellipsoids (min_vol_union_ellip.m)
    • One free point localization (ex_8_5.m)
    • Polynomial discrimination (poly3_discr.m)
    • Polynomial discrimination (poly4_discr.m)
    • Quadratic discrimination (separating ellipsoid) (quad_discr.m)
    • Separating ellipsoids in 2D (separate_ell_2D.m)
    • Solve a floor planning problem given graphs H & V (floor_plan_graphs.m)
    • Utility functions:
      • Computes a minimum-perimeter bounding box subject to positioning constraints (floorplan.m)
• Filter design

  Relevant links: ~boyd/magdes.html class/ee364/lectures/filters.pdf

  • Chebychev design of an FIR filter given a desired H(w) (fir_chebychev_design.m)
  • Design a 1/f spectrum shaping (pink-noise) filter (one_over_f_filter.m)
  • Equalizer design example (equalizer_design.m)
  • Maximize stopband attenuation of a bandpass IIR filter (iir_mag_design_bandpass_max_atten.m)
  • Maximize stopband attenuation of a linear phase lowpass FIR filter (fir_lin_phase_lowpass_max_atten.m)
  • Maximize stopband attenuation of a lowpass FIR filter (magnitude design) (fir_mag_design_lowpass_max_atten.m)
  • Maximize stopband attenuation of a lowpass IIR filter (iir_mag_design_lowpass_max_atten.m)
  • Minimize order of a linear phase lowpass FIR filter (fir_lin_phase_lowpass_min_order.m)
  • Minimize order of a lowpass FIR filter (magnitude design) (fir_mag_design_lowpass_min_order.m)
  • Minimize stopband ripple of a linear phase lowpass FIR filter (fir_lin_phase_lowpass_min_ripple.m)
  • Minimize transition bandwidth of a linear phase lowpass FIR filter (fir_lin_phase_lowpass_min_trans.m)
  • Utility functions:
    • Spectral factorization using Kolmogorov 1939 approach. (spectral_fact.m)
• Geometric programming

  From Boyd, Kim, Vandenberghe, and Hassibi, "A Tutorial on Geometric Programming": ~boyd/gp_tutorial.html

  • Box volume maximization (max_volume_box.m)
  • Digital circuit sizing (simple_dig_ckt_sizing.m)
  • Digital circuit sizing (vectorized) (simple_dig_ckt_sizing_vect.m)
  • Elmore delay sizing for an interconnect network. (elmore_interconnect.m)
  • Floor planning with an optimal trade-off curve. (floor_planning.m)
  • Optimal doping profile optimization (basic_odp.m)
  • Optimal doping profile optimization with current gain constraint. (beta_min_odp.m)
  • Simple power control in communication systems via GP. (power_control.m)
• Graph Laplacian eigenvalue optimization

  From a talk delivered by S. Boyd to the 2006 ICM: ~boyd/cvx_opt_graph_lapl_eigs.html

  • FDLA and FMMC solutions for a 50-node, 200-edge graph (larger_example.m)
  • FDLA and FMMC solutions for a 64-node, 95-edge cut-grid graph (cut_grid_example.m)
  • FDLA and FMMC solutions for an 8-node, 13-edge graph (small_example.m)
  • Utility functions:
    • Computes fastest mixing Markov chain (FMMC) edge weights (fmmc.m)
    • Computes the Metropolis-Hastings heuristic edge weights (mh.m)
    • Computes the constant edge weight that yields fastest averaging. (best_const.m)
    • Computes the fastest distributed linear averaging (FDLA) edge weights (fdla.m)
    • Computes the maximum-degree heuristic edge weights (max_deg.m)
    • Generate a cut-grid graph for the ICM 2006 talk example (cut_grid_data.m)
    • Plots a graph with each edge width proportional to its weight. (plotgraph.m)
• Sparse solution heuristics
  • Computing a sparse solution of a set of linear inequalities (sparse_solution.m)
  • Detecting a small subset of infeasible linear inequalities (sparse_infeas_dual.m)
  • Finding a point that satisfies many linear inequalities (sparse_infeas.m)
• Miscellaneous examples
  • Examples from the CVX User's guide (quickstart.m)
  • Builds a norm minimization tradeoff curve (regularized_norm_tradeoff.m)
  • Builds and solves a simple least-squares problem using cvx (simple_LS.m)
  • Builds and solves a simple linear program (simple_LP.m)
  • Closest Toeplitz SDP search. (closest_toeplitz_psd.m)
  • Equality constrained norm minimization. (equality_constr_norm_min.m)
  • Minimal phase spectral factorization (min_phase_spectral_fact.m)
  • Nonnegative matrix factorization (nonneg_matrix_fact.m)

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