systems op exam 1 true false questions
In the general optimum design problem formulation, the number of inequality constraints cannot exceed the number of design variables.
F (any #)
A feasible design may violate " > or = type" constraints.
F (cannot)
A feasible design may violate equality constraints.
F (cannot)
Maximization of f(x) is equivalent to minimization of 1/f(x).
F (change 1/f(x) to -f(x))
In optimum design problem formulation, " > or = type" constraints cannot be treated.
F (convert to < or =)
At the optimum point, the number of active independent constraints is always more than the number of design variables
F (equal or less)
Let fn be the minimum value for the cost function with n design variables for a problem. If the number of design variables for the same problem is increased to, say, m 5 2n, then fm > fn, where fm is the minimum value for the cost function with m design variables.
F (fm < fn)
All design problems have only linear inequality constraints.
F (get rid of only)
The gradient of a function at a point gives a local direction of maximum decrease in the function.
F (increase)
A function defined on an open set cannot have a global minimum
F (it can have a global minimum)
A lower minimum value for the cost function is obtained if more constraints are added to the problem formulation.
F (less constraints)
If curvature of an unconstrained function of a single variable at the point x* is zero, then it is a local maximum point for the function.
F (maximum not local max)
Design problems with equality constraints have the gradient of the cost function as zero at the optimum point.
F (not necessarily 0)
Gradients of inequality constraints that are active at the optimum point must be zero.
F (not necessary)
The curvature of an unconstrained function of a single variable at its local minimum point is negative
F (positive not negative)
Optimum design points having at least one active constraint give stationary value to the cost function.
F (stationary value to L equation)
A symmetric matrix is positive definite if its eigenvalues are non-negative.
F (strictly positive not non-negative)
If a constant is added to a function, the location of its minimum point is changed.
F (would not change)
If a slack variable has zero value at the optimum, the inequality constraint is inactive.
F (active)
A matrix is positive semidefinite if some of its eigenvalues are negative and others are non-negative.
F (all non-negative)
The Hessian matrix of a continuously differentiable function can be asymmetric.
F (Hessian is always symmetric)
The Hessian matrix for a function is calculated using only the first derivatives of the function.
F (Second derivative)
A regular point of the feasible region is defined as a point where the cost function gradient is independent of the gradients of active constraints.
F
If the Hessian of an unconstrained function is indefinite at a candidate point, the point may be a local maximum or minimum.
F
The Hessian of an unconstrained function at its minimum point is negative definite.
F
A function cannot have more than one global minimum point.
F ( can have more than one)
Taylor series expansion can be written at a point where the function is discontinuous.
F ( must be continuous)
The number of " < or =" constraints must be less than the number of design variables for a valid problem formulation
F ( no limit on inequality constraints)
A quadratic form can have first-order terms in the variables.
F ( only allowed to have second-order terms)
The number of independent equality constraints can be larger than the number of design variables for the problem
F (# of Design Var. has to be < or =)
For a given x, the quadratic form defines a vector.
F (F(x) not x)
A "< or = type" constraint expressed in the standard form is active at a design point if it has zero value there
T
A function can have a negative value at its maximum point.
T
A function can have several local minimum points in a small neighborhood of x*
T
A point satisfying KKT conditions for a general optimum design problem can be a local max-point for the cost function.
T
A point satisfying first-order necessary conditions for an unconstrained function may not be a local minimum point.
T
A positive definite quadratic form must have positive value for any x NOT= 0.
T
All design variables should be independent of each other as far as possible.
T
All eigenvalues of a negative definite matrix are strictly negative.
T
At a constrained optimum design point that is regular, the cost function gradient is linearly dependent on the gradients of the active constraint functions.
T
At the optimum point, Lagrange multipliers for the " < or = type" inequality constraints must be non-negative.
T
At the optimum point, the Lagrange multiplier for a " < or = type" constraint can be zero.
T
Design of a system implies specification of the design variable values.
T
Each optimization problem must have certain parameters called the design variables.
T
Every quadratic form has a symmetric matrix associated with it.
T
If a function is multiplied by a positive constant, the location of the function's minimum point is unchanged.
T
If the first-order necessary condition at a point is satisfied for an unconstrained problem, it can be a local maximum point for the function.
T
If there is an equality constraint in the design problem, the optimum solution must satisfy it
T
In the general optimum design problem formulation, the number of independent equality constraints must be " < or = " to the number of design variables.
T
Linear Taylor series expansion of a complicated function at a point is only a good local approximation for the function.
T
Optimum design points for constrained optimization problems give stationary value to the Lagrange function with respect to design variables.
T
Taylor series expansion for a function at a point uses the function value and its derivatives
T
Taylor series expansion of a complicated function replaces it with a polynomial function at the point.
T
The Hessian of an unconstrained function at its local minimum point must be positive semidefinite.
T
The constraint set for a design problem consists of all feasible points.
T
The feasible region for an equality constraint is a subset of that for the same constraint expressed as an inequality.
T
The gradient of a function f(x) at a point is normal to the surface defined by the level surface f(x) 5 constant.
T
The quadratic form appears as one of the terms in Taylor's expansion of a function.
T
The value of the function having a global minimum at several points must be the same.
T
While solving an optimum design problem by KKT conditions, each case defined by the switching conditions can have multiple solutions.
T
