% choose one of: 'interior-point', 'sqp', 'active-set', 'trust-region-reflective' 'AlwaysHonorConstraints', 'bounds'. Max(constraint violation) = 2.44e-11 ConstraintTolerance = 1e-06 (selected)Īctive inequalities (to within options.ConstraintTolerance = 1e-06):įunction = optimize () % - starting point and bounds - x0 = ub = % upper bound, the largest values we will allow x to acheive lb = % lower bound % - % - linear constraints - A = b = Aeq = beq = % - % - Objective and Constraints - function = objcon (x ) % set objective/constraints here % objective J = ( 1 - x ( 1 ))^ 2 100 * ( x ( 2 ) - x ( 1 )^ 2 )^ 2 % inequality constraints c = zeros ( 2, 1 ) % there are 2 of them c ( 1 ) = x ( 1 )^ 2 x ( 2 )^ 2 - 1 c ( 2 ) = x ( 1 ) 3 * x ( 2 ) - 5 % equality constraints ceq = % there are none end % - % - options - options = optimoptions fmincon. Violation, 2.436318e-11, is less than options.ConstraintTolerance = 1.000000e-06.Ībs(steplength*directional derivative) = 5.35e-07 FunctionTolerance = 1e-06 (selected) Optimization stopped because the predicted change in the objective function,ĥ.349514e-07, is less than options.FunctionTolerance = 1.000000e-06, and the maximum constraint Number of linear inequality constraints: 0 ![]() Number of nonlinear equality constraints: 0 Number of nonlinear inequality constraints: 4 Nonlinear constraints gradient: finite-differencing Hessian: finite-differencing (or Quasi-Newton) ![]() In short, we can write all of our optimization problems in this form: Without these constraints, our spoiler may produce a lot of lift but may snap in half or be too difficult to manufacture. In our car example the constraints may be things like a constraint on bending stress, manufacturing tolerances, etc. This is completely general as a constraint of the form $q(x) \ge 0$ could be rewritten as $c(x) = -q(x) \le 0$. By convention we will define constraints in this form: $c(x) \le 0$. We denote them with $c$, where $c$ is an array because we may have many constraints. In our car example, the objective of our spoiler design may be to maximize (downward) lift.Ĭonstraints exist in almost any engineering problem, they are conditions that we require the optimizer to satisfy. ![]() This does not mean that the algorithms we explore can can only perform minimization, to maximize you simply minimize the negative of the function ($-J(x)$). By convention we say minimize because we often minimize things in engineering, but in other disciplines, like finance, the default convention may be to maximize. It is a scalar, meaning a single value (at times it makes sense to have multiple objectives, but the approach is different-we will explore this later in the semester). We usually denote this function by $J$ or sometimes $f$. The objective is the metric we want to minimize. For example if we were designing the shape of the spoiler on a car, the design variables might be things like the length of the spoiler, the distance away from the car body, and parameters that define its curved profile. We denote these variables as $x$, where $x$ is an array (i.e., there can be many design variables). An optimization problem has three main elements:ĭesign variables are the variables that we allow the optimizer to change.
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