--- title: "Notes on LP, GP and multivariable search" subject: "Machine Design II" chapter: 2 key: "semester_7/machine_design_ii:lp-geometric-programming-notes" source: https://www.bctengineer.com/q/semester_7/machine_design_ii/lp-geometric-programming-notes site: BCT Engineering --- # Notes on LP, GP and multivariable search Machine Design II · Chapter 2: Optimization Techniques - Asked in: year not given, 4+4 marks - Video: [Linear Programming (Optimization) 2 Examples Minimize & Maximize](https://www.youtube.com/watch?v=Y7e7DCsDUMY) — Mario's Math Tutoring - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_7/machine_design_ii/lp-geometric-programming-notes - Whole chapter: https://www.bctengineer.com/md/semester_7/machine_design_ii/chapter-2 ## Question Write short notes on: (a) linear programming and geometric programming; (b) multivariable search methods and multifactor objective functions in design optimisation. ## Short answer Linear programming optimizes a linear objective under linear constraints, with optimum at a vertex, solved by simplex. Geometric programming deals with posynomial objectives via dual weights. Multivariable search includes univariate, steepest descent and pattern methods. Multifactor objectives combine by weighted sums or constraints, generating a Pareto set. ## Answer #### (a) Linear programming and geometric programming **Linear programming (LP)** optimises a linear objective subject to linear constraints. $$ \text{Maximise } Z = c_1x_1 + c_2x_2 + \dots \quad \text{subject to } \sum a_{ij}x_j \le b_i,\ x_j \ge 0 $$ - Graphical method for two variables: plot constraints, find the feasible region, and check the corner points. The optimum is always at a vertex. - For many variables the **simplex method** is used. - Design use: product mix, material cutting, allocation of machine hours. **Geometric programming (GP)** handles objective and constraints written as **posynomials** (sums of positive terms with product of powers): $$ f(x) = \sum_k c_k\, x_1^{a_{k1}} x_2^{a_{k2}} \cdots x_n^{a_{kn}},\ c_k>0 $$ Using the arithmetic-geometric mean inequality it gives the optimum cost from the **dual** problem, often before the design variables are known. The weights satisfy normality $\sum w_k = 1$ and orthogonality conditions. The number of degrees of difficulty is (number of terms) $-$ (variables) $-$ 1; zero means the solution is found by solving linear equations. Use: design of tanks, beams, gear trains where cost has power-law terms. #### (b) Multivariable search methods Methods for several variables without derivatives (or with them): - **Lattice (grid) search:** evaluates a grid; simple but expensive. - **Univariate search:** change one variable at a time to its best value, then the next, and repeat. - **Steepest descent (gradient):** move along $-\nabla f$ with a step length found by line search. - **Pattern search (Hooke-Jeeves), Simplex (Nelder-Mead), Newton and conjugate gradient** for faster convergence. ```text x2 | . . . contour of f | . * . * = optimum | . / . steps follow | /___. descent path +----------- x1 ``` #### Multifactor objective functions Real designs have several goals (weight, cost, life, safety). Methods to combine them: - **Weighted sum:** $F = w_1 f_1 + w_2 f_2 + \dots$ with $\sum w_i = 1$ (after normalising the $f_i$). - **Constraint method:** optimise the most important goal and convert the others to constraints. - **Pareto set:** collection of designs where one objective cannot improve without worsening another; the designer chooses from it. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In the graphical method for linear programming, where does the optimum lie? A. At a vertex (corner point) of the feasible region B. At the centre of the feasible region C. On any point of an edge only D. Outside the feasible region Answer: A. With a linear objective and linear constraints the optimum is always at a vertex of the feasible region. (Key point: LP optimum location) 2. What kind of functions does geometric programming handle? A. Only linear functions B. Posynomials, sums of positive-coefficient terms with products of powers of the variables C. Only functions with negative coefficients D. Only trigonometric functions Answer: B. GP is written for posynomials with c_k > 0 and the arithmetic-geometric mean inequality is used. (Key point: Posynomials) 3. How are the degrees of difficulty of a geometric programme found, and what does zero mean? A. Variables minus terms; zero means no solution B. Terms times variables; zero means unbounded C. Terms minus variables minus 1; zero means the weights come from linear equations D. Constraints minus 1; zero means a graphical solution Answer: C. Degrees of difficulty = terms - variables - 1. At zero, normality and orthogonality conditions give the weights by linear equations. (Key point: Degrees of difficulty) 4. What is a Pareto set in multi-objective design? A. The design with the lowest cost only B. A set of designs with equal weights C. The designs that violate all constraints D. Designs where no objective can be improved without worsening another Answer: D. Pareto designs are non-dominated trade-offs, and the designer chooses among them. 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