1. Problem Formulation

General Structure Objective Function: The objective function is usually formulated on the basis of economic criterion, e.g. profit, cost, energy and yield, etc., as a function of key variables of the system under study. Process Model: They are used to describe the interrelations of the key variables.

Example – Thickness of Insulation

Essential Features of Optimization Problems At least one objective function, usually an economic model; Equality constraints; Inequality constraints. Categories 2 and 3 are mathematical formulations of the process model. A feasible solution satisfies both the equality and inequality constraints, while an optimal solution is a feasible solution that optimize the objective function.

Mathematical Notation Objective function Equality constraints Inequality constraints

Economic Objective Function = income - operating costs - capital costs

EXAMPLE: OPTIMUM THICKNESS OF INSULATION

The insulation has a lifetime of 5 years. The fund to purchase and install the insulation can be borrowed from a bank and paid back in 5 annual installments. Let r be the fraction of the installed cost to be paid each year to the bank. (r>0.2)

Time Value of Money The economic analysis of projects that incur income and expense over time should include the concept of the time value of money. This concept means that a unit of money on hand NOW is worth more than the same unit of money in the future.

Investment Time Line Diagram

Example You deposit $1000 now (the present value P) in a bank saving account that pays 5% annual interest compounded monthly. You plan to deposit $100 per month at the end of month for the next year What will the future value F of your investment be at the end of next year?

Present Value and Future Worth

Present Value of a Series of (not Necessarily Equal) Payments

Present Value of a Series of Uniform Future Payments

Repayment Multiplier

Future Value of a Series of (not Necessarily Equal) Payments

Future Value of a Series of Uniform Future Payments

Measures of Profitability

Measures of Profitability Net present value (NPV) is calculated by adding the initial investment (represented as a negative cash flow) to the present value of the anticipated future positive (and negative) cash flows. Internal rate of return (IRR) is the rate of return (i.e. interest rate or discount rate) at which the future cash flows (positive plus negative) would equal the initial cash outlay (a negative cash flow).

2. Basic Concepts

Continuity of Functions

Continuity of Functions

Stationary Point

Unimodal and Multimodal Functions A unimodal function has one extremum. A multimodal function has more than one extrema. A global extremum is the biggest (or smallest) among a set of extrema. A local extremum is just one of the extrema.

Two extrema: f=3 f=7

Two extrema: a and b

Definition of Unimodal Function

Convex and Concave Single-Variable Functions A function is called convex over a region R, if, for any two values of x in R, the following inequality holds

Hessian Matrix

Hessian Matrix

Positive and Negative Definiteness

Remarks A function is convex (strictly convex) iff its Hessian matrix is positive semi-definite (definite). A function is concave (strictly concave) iff its Hessian matrix is negative semi-definite (definite).

Tests for Strictly Convexity All diagonal elements of Hessian matrix must be positive. Also, the determinants of Hessian matrix and all its leading principal minors must all be positive. All eigenvalues of Hessian matrix must be positive.

Convex Region

Convex Region

Why do we need to discuss convexity and concavity? Determination of convexity or concavity can be used to establish whether a local optimal solution is also a global optimal solution. If the objective function is known to be convex or concave, computation of optimum can be accelerated by using appropriate algorithm.

Convex Programming Problem

A NLP is generally not a convex programming problem!

Proposition

Linear Varieties

Hyperplanes

Half Spaces Clearly the half spaces are convex sets.

Polytope and Polyhedron A set which can be expressed as the intersection of a finite number of closed half spaces is said to be a convex polytope. A nonempty bounded polytope is called a polyhedron.

Necessary Conditions for the Extremum of an Unconstrained Function Implies one of the following three possibilities: (1) a minimum, (2) a maximum or (3) a saddle point.

Sufficient Conditions for the Extremum of an Unconstrained Function

Theorem Suppose at a point the first n-1 derivatives are zero and the first nonzero higher-order derivative is denoted by n, If n is odd, this point is a inflection point. If n is even, then it is a local extremum. Moreover,