Decision Making . . . What If the Problem . . . How to Describe Final . . . Describing Final . . . Why Convex Optimization Need to Consider . . . Is Ubiquitous and Why Main Result Decision Making . . . Pessimism Is Widely Spread When Is This Convex? This Explains Why . . . Angel F. Garcia Contreras, Home Page Martine Ceberio, and Vladik Kreinovich Title Page Department of Computer Science ◭◭ ◮◮ University of Texas at El Paso ◭ ◮ 500 W. University El Paso, Texas 79968, USA Page 1 of 19 afgarciacontreras@miners.utep.edu, mceberio@utep.edu@utep.edu, vladik@utep.edu Go Back Full Screen Close Quit
Decision Making . . . What If the Problem . . . 1. Decision Making Means Optimization How to Describe Final . . . • In many real life situations, we need to make a decision, Describing Final . . . i.e., select an alternative x out of many. Need to Consider . . . Main Result • Decision making theory has shown that: Decision Making . . . – the decision making of a rational person When Is This Convex? – is equivalent to maximizing a special function u ( x ) This Explains Why . . . ( utility ) that describes this person’s preferences. Home Page • Thus, maximization problems are very important for Title Page practical applications. ◭◭ ◮◮ • In many cases, the utility value is described by its mon- ◭ ◮ etary equivalent amount. Page 2 of 19 • Small changes in an alternative should lead to small Go Back change in preferences, so u ( x ) is continuous. Full Screen Close Quit
Decision Making . . . What If the Problem . . . 2. What If the Problem has Several Solutions? How to Describe Final . . . • The optimization problem can have several solutions: Describing Final . . . Need to Consider . . . u ( x (1) ) = u ( x (2) ) = . . . = max u ( x ) . Main Result x Decision Making . . . • From the practical viewpoint, we can use this non- When Is This Convex? uniqueness to optimize something else. This Explains Why . . . • E.g., if several designs x (1) , x (2) , . . . are equally prof- Home Page itable, we select the most environmentally friendly one. Title Page • If we still have several possible alternatives, we can, ◭◭ ◮◮ e.g., look for the most aesthetically pleasing design. ◭ ◮ • This process continues until we end up with the single Page 3 of 19 optimal alternative. Go Back • So, the final objective function should have the unique maximum. Full Screen Close Quit
Decision Making . . . What If the Problem . . . 3. How to Describe Final Objective Functions? How to Describe Final . . . • In general, selecting a decision x involves selecting the Describing Final . . . values of many different parameters x 1 , . . . , x n . Need to Consider . . . Main Result • For example, when we select a design of a plant, we Decision Making . . . must take into account: When Is This Convex? – the land area that we need to purchase, This Explains Why . . . – the amount of steel and concrete that goes into con- Home Page struction, Title Page – the overall length of roads, pipes, etc. forming the ◭◭ ◮◮ supporting infrastructure, etc. ◭ ◮ • Our original decision x is based on known costs of all Page 4 of 19 these attributes. Go Back • However, costs can change. Full Screen Close Quit
Decision Making . . . What If the Problem . . . 4. Describing Final Objective Functions (cont-d) How to Describe Final . . . • If the cost per unit of the i -th attribute changes by the Describing Final . . . value d i , then the overall cost of x changes to Need to Consider . . . Main Result n � u ′ ( x ) = u ( x ) + d i · x i . Decision Making . . . i =1 When Is This Convex? This Explains Why . . . • It is therefore reasonable to select an objective function Home Page u ( x ) in such away that: Title Page – for all possible combinations of values d i , ◭◭ ◮◮ – the resulting combination also has the unique max- imum. ◭ ◮ Page 5 of 19 Go Back Full Screen Close Quit
Decision Making . . . What If the Problem . . . 5. Need to Consider Constraints How to Describe Final . . . • In practice, there are always physical and economical Describing Final . . . restrictions on the possible values of these parameters. Need to Consider . . . Main Result • As a result, for each parameter x i , we always have Decision Making . . . bounds x i and x i , so x i ∈ [ x i , x i ]. When Is This Convex? • Under such constraints, the optimization problem al- This Explains Why . . . ways has a solution Home Page • Indeed, on a bounded closed set B = [ x 1 , x 1 ] × . . . × Title Page [ x n , x n ], every continuous u ( x ) attaints its maximum. ◭◭ ◮◮ ◭ ◮ Page 6 of 19 Go Back Full Screen Close Quit
Decision Making . . . What If the Problem . . . 6. Definition and Discussion How to Describe Final . . . • A continuous function u ( x ) = u ( x 1 , . . . , x n ) is called a Describing Final . . . final objective function if: Need to Consider . . . Main Result – for every combination of tuples d = ( d 1 , . . . , d n ), Decision Making . . . x = ( x 1 , . . . , x n ), and x = ( x 1 , . . . , x n ) When Is This Convex? – the following constrained optimization problem has This Explains Why . . . the unique solution: Home Page n � Maximize u ( x ) + d i · x i under constraints Title Page i =1 x i ≤ x i ≤ x i . ◭◭ ◮◮ • This is true for strictly convex functions u ( x ), for which ◭ ◮ � x + x ′ > u ( x ) + u ( x ′ ) � for all x � = x ′ . u Page 7 of 19 2 2 Go Back Full Screen Close Quit
Decision Making . . . What If the Problem . . . 7. Discussion (cont-d) How to Describe Final . . . • Indeed, it is easy to prove that for a strictly convex Describing Final . . . function, maximum is attained at a unique point: Need to Consider . . . – if we have two different points x � = x ′ at which Main Result u ( x ) = u ( x ′ ) = max Decision Making . . . u ( x ), x When Is This Convex? – then, due to strong convexity, for the midpoint = x + x ′ This Explains Why . . . x ′′ def , we would have u ( x ′′ ) > u ( x ) = u ( x ′ ); Home Page 2 – this would imply u ( x ′′ ) > max u ( x ), which is not Title Page x possible. ◭◭ ◮◮ • If u ( x ) is strictly convex, it remains strictly convex ◭ ◮ n after adding � d i · x i . Page 8 of 19 i =1 Go Back • Thus, strictly convex functions are indeed final objec- tive functions. Full Screen • Interestingly, they are the only ones. Close Quit
Decision Making . . . What If the Problem . . . 8. Main Result How to Describe Final . . . • Proposition. Every smooth final objective function Describing Final . . . u ( x ) is convex. Need to Consider . . . Main Result • This result explains why convex objective functions are Decision Making . . . ubiquitous in practical applications. When Is This Convex? • This result is also good for practical applications since: This Explains Why . . . Home Page – while optimization in general is NP-hard, – feasible algorithms are known for solving convex Title Page optimization problem. ◭◭ ◮◮ ◭ ◮ Page 9 of 19 Go Back Full Screen Close Quit
Decision Making . . . What If the Problem . . . 9. Decision Making Under Uncertainty How to Describe Final . . . • In many practical situations, we do not know the exact Describing Final . . . consequences of different actions. Need to Consider . . . Main Result • So, for each alternative x , we have several different Decision Making . . . values u ( x, s ) depending on the situation s . When Is This Convex? • According to decision theory, a reasonable idea is to This Explains Why . . . optimize the so-called Hurwicz criterion Home Page U ( x ) = α · max u ( x, s )+(1 − α ) · min u ( x, s ) for some α ∈ [0 , 1] . Title Page s s ◭◭ ◮◮ • Here, α = 1 corresponds to the optimistic approach, when we only consider the best-case scenarios. ◭ ◮ • α = 0 is pessimistic approach, when we only consider Page 10 of 19 the worst cases. Go Back • α ∈ (0 , 1) means that we consider both the best and Full Screen the worst cases. Close Quit
Decision Making . . . What If the Problem . . . 10. When Is This Convex? How to Describe Final . . . • We showed that we should consider situations in which: Describing Final . . . Need to Consider . . . • u ( x, s ) is convex for every s and Main Result • the objective function U ( x ) is also convex. Decision Making . . . • For α = 0, it is easy to show that the minimum of When Is This Convex? convex function is always convex. This Explains Why . . . Home Page • For α = 0 . 5, we get arithmetic average – also convex. Title Page • Case α < 0 . 5 is a convex combination of α = 0 and ◭◭ ◮◮ α = 0 . 5, so also convex. ◭ ◮ • However, for α > 0 . 5, this is no longer true: Page 11 of 19 • E.g., for u ( x, +) = | x − 1 | and u ( x, − ) = | x + 1 | , the function U ( x ) attains maximum for two different x . Go Back Full Screen • Thus, U ( x ) is not convex. Close Quit
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