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Check which MATLAB toolboxes I have installed, specifically looking for Optimization Toolbox and Global Optimization Toolbox.
You have both toolboxes installed:
| Toolbox | Version |
|---|---|
| Optimization Toolbox | 25.2 (R2025b) |
| Global Optimization Toolbox | 25.2 (R2025b) |
You're running MATLAB R2025b with a comprehensive installation that includes 100+ toolboxes. Both optimization toolboxes are available and ready to use for your optimization work.
I need to optimize a cantilever beam design. The beam has length L=5m, carries a tip load P=10kN, made of steel with E=200 GPa and yield stress=250 MPa. I want to minimize the cross-sectional area (width × height) while keeping tip deflection under 10mm and max stress under the yield stress. Width should be between 50-200mm, height between 100-500mm. Please set this up and solve it using MATLAB.
The optimization completed successfully. Here's a summary of the results:
| Parameter | Value |
|---|---|
| Width (b) | 50.00 mm |
| Height (h) | 368.40 mm |
| Cross-sectional Area | 18,420 mm² |
| Constraint | Value | Limit | Status |
|---|---|---|---|
| Tip Deflection | 10.00 mm | 10.0 mm | Active (at limit) |
| Max Stress | 44.21 MPa | 250 MPa | Satisfied |
Deflection-controlled design: The deflection constraint is the active constraint (exactly at the 10mm limit), while the stress constraint has significant margin (safety factor of 5.66).
Width at lower bound: The optimizer pushed the width to its minimum (50mm) because for deflection-controlled beams, it's more efficient to increase height than width. The moment of inertia depends on h³, so increasing height is much more effective at reducing deflection.
Convergence: The SQP algorithm converged in just 5 iterations with 18 function evaluations, achieving first-order optimality.
The visualization in MATLAB shows: - The optimal rectangular cross-section - The deflection curve along the beam length - A contour plot of the feasible design space with constraint boundaries
If you need a more conservative design with margin on the deflection constraint, or want to explore different cross-section shapes (I-beam, hollow sections), let me know.
I see you've selected fmincon for this particular problem. Please explain why this is the right solver for this kind of problem
Great question. Let me explain why fmincon is well-suited for this problem by analyzing the problem characteristics.
1. Continuous Design Variables - Width (b) and height (h) are continuous variables within bounded ranges - Not discrete/integer choices
2. Nonlinear Objective Function - Area = b × h is bilinear (product of two variables) - Not a linear function of the design variables
3. Nonlinear Inequality Constraints - Deflection: δ = 4PL³/(Ebh³) ≤ δ_max — involves h³ in denominator - Stress: σ = 6PL/(bh²) ≤ σ_y — involves h² in denominator - Both constraints are nonlinear functions of the design variables
4. Bound Constraints - Simple box constraints on b and h
5. Smooth Functions - All functions are continuously differentiable - No discontinuities or non-smooth behavior
| Requirement | fmincon Capability |
|---|---|
| Continuous variables | ✓ Designed for continuous optimization |
| Nonlinear objective | ✓ Handles general nonlinear objectives |
| Nonlinear constraints | ✓ Supports nonlinear inequality/equality constraints |
| Bound constraints | ✓ Native support via lb and ub |
| Smooth functions | ✓ Exploits gradient information (SQP algorithm) |
| Local minimum | ✓ Guaranteed convergence to local optimum |
| Solver | Why Not Ideal |
|---|---|
linprog |
Only for linear objectives and constraints |
quadprog |
Only for quadratic objectives with linear constraints |
fminunc |
No constraint handling capability |
ga (genetic algorithm) |
Overkill for smooth problems; slower convergence |
patternsearch |
Derivative-free; less efficient for smooth problems |
fsolve |
For equation solving, not optimization |
I selected the Sequential Quadratic Programming (SQP) algorithm within fmincon because:
For this problem, fmincon with SQP is essentially the textbook choice — it's the standard tool for smooth, constrained, nonlinear optimization in MATLAB.
Use the problem-based skill to solve this instead
The problem-based approach produces the same optimal solution but with a much cleaner formulation. Here's a comparison:
| Aspect | Solver-Based (fmincon) | Problem-Based |
|---|---|---|
| Variable definition | Implicit in vector x |
Explicit: optimvar('b', ...) |
| Objective | Anonymous function @(x) x(1)*x(2) |
Direct expression: b * h |
| Constraints | Separate function returning [c, ceq] |
Readable: 4*P*L^3 <= delta_max * E * b * h^3 |
| Bounds | Separate lb, ub vectors |
Built into optimvar() |
| Readability | Requires understanding indexing | Self-documenting, math-like syntax |
1. Readable Formulation
% Problem-based - reads like the actual constraint
prob.Constraints.deflection = 4*P*L^3 <= delta_max * E * b * h^3;
% vs Solver-based - requires separate function
c1 = (P * L^3) / (3 * E * I) - delta_max;
2. Named Variables
b = optimvar('b', 'LowerBound', 0.05, 'UpperBound', 0.20);
h = optimvar('h', 'LowerBound', 0.10, 'UpperBound', 0.50);
% vs x(1), x(2) with separate lb/ub vectors
3. Self-Documenting Output
The show(prob) command displays the complete problem formulation in a human-readable format, making it easy to verify the setup.
4. Automatic Solver Selection
MATLAB analyzes the problem structure and selects an appropriate solver (it chose fmincon here since constraints are nonlinear).
5. Named Solution Access
sol.b, sol.h % vs x_opt(1), x_opt(2)
The problem-based approach is generally preferred for new code as it's more maintainable and less error-prone, especially as problems grow in complexity.