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