Optimization-Overview优化设计.pptVIP

Active constraint – an inequality constraint gi(x)≤0 is said to be active at a design point x* if it is satisfied at equality, i.e., gi(x*)=0. All equality constraints are active for all feasible designs An inequality constraint gi(x) ≤0 is said to be inactive at a design point x* if it is strictly satisfied, i.e. gi(x*)0. An inequality constraint gi(x) ≤0 is said to be violated at a design point x* if its value is positive, i.e., gi(x*)0. An equality constraint hi(x)=0 is violated at a design point x* if hi(x*) is not identically zero. Example Problem with multiple solutions When a constraint is parallel to the cost function and if the constraint is active at the optimum, then there are multiple solutions for the problem Minimize f=-x1-0.5x2 Subject to: 2x1 + 3x2 ≤ 12 2x1 + x2 ≤ 8 -x1 ≤ 0 -x2 ≤ 0 Optimum: f=-4 Optimum points: any points along the line A-B A B Example Problem with unbounded solution Maximize f=x1-2x2 Subject to: 2x1 - x2 ≥ 0 -2x1 + 3x2 ≤ 6 x1 ≥ 0 x2 ≥ 0 Standard form Minimize f’=-x1+2x2 Subject to: -2x1 + x2 ≤ 0 -2x1 + 3x2 ≤ 6 -x1 ≤ 0 -x2 ≤ 0 Example Infeasible problem Minimize f=x1+2x2 Subject to: 3x1 + 2x2 ≤ 6 2x1 + 3x2 ≤ 12 x1 ≤ 5 x2 ≤ 5 -x1 ≤ 0 -x2 ≤ 0 Example Minimum weight tubular column Straight columns as structural elements are used in many civil, mechanical, aerospace and automotive structures. The problem is to design a minimum weight tubular column of length l supporting a load P without buckling or overstressing. The column is fixed at the base and free at the top. Buckling load for such a column is given as . Here I is the moment of inertia for the cross-section of the column and E is the modulus of elasticity. Formulate the design problem and solve graphically. Given: P=10MN,E=207GPa, ?=7833kg/m3, l=5.0m, σa=248MPa assume Rt, A= 2πRt, I= πR3t Find: R (mean radius), t (wall thickness) Minimize: f=mass=?lA= 2?lπRt Subject to: σ ≤ σa buckl