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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.
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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.
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 Objective function = income - operating costs - capital costs
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.
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 of a Series of (not Necessarily Equal) Payments
Future Value of a Series of (not Necessarily Equal) Payments
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).
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
Convex and Concave Single-Variable Functions A function is called convex over a region R, if,for any two values of x inR, the following inequality holds
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.
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.
