In Science, Simplicity . . . In Practice, Simplified . . . Simplified . . . Question Simplicity Is Worse Than What Simplification . . . Theft: A Constraint-Based The Simplified . . . Optimization: Case of . . . Explanation of a Seemingly Optimization: Case of . . . For Optimization, the . . . Counter-Intuitive Russian Home Page Saying Title Page ◭◭ ◮◮ Martine Ceberio, Olga Kosheleva, ◭ ◮ and Vladik Kreinovich Page 1 of 14 University of Texas at El Paso El Paso, TX 79968, USA Go Back mceberio@utep.edu, olgak@utep.edu, Full Screen vladik@utep.edu Close Quit
In Science, Simplicity . . . In Practice, Simplified . . . 1. In Science, Simplicity Is Good Simplified . . . • The world around us is very complex. Question What Simplification . . . • One of the main objectives of science is to simplify it. The Simplified . . . • Science has indeed greatly succeeded in doing it. Optimization: Case of . . . • Example: Newton’s equations explain the complex mo- Optimization: Case of . . . tions of celestial bodies motion by simple laws. For Optimization, the . . . Home Page • From this viewpoint, simplicity of the description is Title Page desirable. ◭◭ ◮◮ • To achieve this simplicity, we sometimes ignore minor factors. ◭ ◮ Page 2 of 14 • Example: Newton treated planets as points, while they have finite size. Go Back • As a result, there is a small discrepancy between New- Full Screen ton’s theory and observations. Close Quit
In Science, Simplicity . . . In Practice, Simplified . . . 2. In Practice, Simplified Models Are Not Always Simplified . . . Good in Decision Making Question • One of the main purposes of science is to gain knowl- What Simplification . . . edge. The Simplified . . . Optimization: Case of . . . • Once this knowledge is gained, we use it to improve Optimization: Case of . . . the world; examples: For Optimization, the . . . – knowing how cracks propagate helps design more Home Page stable constructions. Title Page – knowing the life cycle of viruses helps cure diseases ◭◭ ◮◮ caused by these viruses. ◭ ◮ • What happens sometimes is that the simplified models, Page 3 of 14 – models which have led to very accurate predictions , Go Back – are not as efficient when we use them in decision making . Full Screen Close Quit
In Science, Simplicity . . . In Practice, Simplified . . . 3. Simplified Approximate Models Leads to Bad Simplified . . . Decisions: Examples Question • Numerous examples can be found in the Soviet exper- What Simplification . . . iment with the global planning of economy. The Simplified . . . Optimization: Case of . . . • Good ideas: Nobelist Wassily Leontieff started his re- Optimization: Case of . . . search as a leading USSR economist. For Optimization, the . . . • However, the results were sometimes not so good . Home Page • Example: buckwheat – which many Russian like to eat Title Page – was often difficult to buy. ◭◭ ◮◮ • Explanation: to solve a complex optimization problem, ◭ ◮ we need to simplify the problem. Page 4 of 14 • How to simplify: similar quantities (e.g., all grains) are Go Back grouped together. Full Screen Close Quit
In Science, Simplicity . . . In Practice, Simplified . . . 4. Examples (cont-d) Simplified . . . • Reminder: all grains are grouped together. Question What Simplification . . . • Problem: we get slightly less buckwheat per area than The Simplified . . . wheat. Optimization: Case of . . . • So, to optimize grain production, we replace all buck- Optimization: Case of . . . wheat with wheat. For Optimization, the . . . Home Page • Example: optimizing transportation. Title Page • When trucks are stuck in traffic or under-loaded, we decrease tonne-kilometers . ◭◭ ◮◮ • At first glance: maximizing tonne-kilometers is a good ◭ ◮ objective. Page 5 of 14 • “Optimal” plan: fully-loaded trucks circling Moscow :( Go Back • General saying: Simplicity is worse than theft. Full Screen Close Quit
In Science, Simplicity . . . In Practice, Simplified . . . 5. Question Simplified . . . • There is an anecdotal evidence of situations in which: Question What Simplification . . . – the use of simplified models in optimization The Simplified . . . – leads to absurd solutions. Optimization: Case of . . . • How frequent are such situations? Are they typical or Optimization: Case of . . . rare? For Optimization, the . . . Home Page • To answer this question, let us analyze this question from the mathematical viewpoint. Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 14 Go Back Full Screen Close Quit
In Science, Simplicity . . . In Practice, Simplified . . . 6. Reformulating the Question in Precise Terms Simplified . . . • In a general decision making problem: Question What Simplification . . . – we have a finite amount of resources, and The Simplified . . . – we need to distribute them between n possible tasks, Optimization: Case of . . . so as to maximize the resulting outcomes. Optimization: Case of . . . • Examples: For Optimization, the . . . Home Page – a farmer allocates money to different crops, to max- imize profits; Title Page – a city allocates police to different districts, to min- ◭◭ ◮◮ imize crime. ◭ ◮ • For simplicity, assume that all resources are of one Page 7 of 14 type. Go Back • We must distribute x 0 resources between n tasks, i.e., n Full Screen find x 1 , . . . , x n ≥ 0 such that � x i = x 0 . i =1 Close Quit
In Science, Simplicity . . . In Practice, Simplified . . . 7. Reformulating the Question in Precise Terms Simplified . . . (cont-d) Question • We must distribute x 0 resources between n tasks, i.e., What Simplification . . . n The Simplified . . . find x 1 , . . . , x n ≥ 0 such that � x i = x 0 . i =1 Optimization: Case of . . . • In many practical problems, the amount of resources Optimization: Case of . . . is reasonably small. For Optimization, the . . . Home Page • So, we can safely linearize the objective function: Title Page n � f ( x 1 , . . . , x n ) ≈ c 0 + c i · x i . ◭◭ ◮◮ i =1 ◭ ◮ • So, the problem is: Page 8 of 14 n – maximize c 0 + � c i · x i Go Back i =1 n Full Screen � – under the constraint x i = x 0 . i =1 Close Quit
In Science, Simplicity . . . In Practice, Simplified . . . 8. What Simplification Means in This Formula- Simplified . . . tion Question • Simplification means that we replace variables x i with What Simplification . . . close values c i with their sum. The Simplified . . . Optimization: Case of . . . • Let us assume that for all other variables x k , we have Optimization: Case of . . . already selected some values. For Optimization, the . . . • Then, the problem is distributing the remaining re- Home Page sources X 0 to remaining tasks x 1 , . . . , x m . Title Page • The original problem is to maximize the sum f ( x 1 , . . . , x m ) = ◭◭ ◮◮ m m � c i · x i under the constraint � x i = X 0 . ◭ ◮ i =1 i =1 • The simplified problem is to maximize s ( x 1 , . . . , x m ) = Page 9 of 14 m m � � c · x i under the constraint x i = X 0 . Go Back i =1 i =1 Full Screen Close Quit
In Science, Simplicity . . . In Practice, Simplified . . . 9. The Simplified Description Provides a Reason- Simplified . . . able Estimate for the Objective Function Question def • The approximation error a = f ( x 1 , . . . , x m ) − s ( x 1 , . . . , x m ) What Simplification . . . m def The Simplified . . . is a = � ∆ c i · x i , where ∆ c i = c i − c . Optimization: Case of . . . i =1 • Let’s assume that ∆ c i are i.i.d., w/mean 0 and st. dev. σ . Optimization: Case of . . . For Optimization, the . . . � m Home Page � x 2 • Thus, a has mean 0 and st. dev. σ [ a ] = σ · i . i =1 Title Page • When resources are ≈ equally distributed x i ≈ X 0 ◭◭ ◮◮ m , we m get σ [ a ] = X 0 · σ ◭ ◮ � √ m and s ( x 1 , . . . , x m ) = c · x i = c · X 0 . Page 10 of 14 i =1 • Thus, the relative inaccuracy of approximating f by s Go Back is σ [ a ] σ = c · √ m ; it is small when m is large. Full Screen s Close Quit
In Science, Simplicity . . . In Practice, Simplified . . . 10. Optimization: Case of the Original Objective Simplified . . . Function Question • The original problem is to maximize the sum f ( x 1 , . . . , x m ) = What Simplification . . . m m The Simplified . . . � � c i · x i under the constraint x i = X 0 . i =1 i =1 Optimization: Case of . . . • From the mathematical viewpoint, this optimization Optimization: Case of . . . problem is easy to solve: For Optimization, the . . . Home Page m � – to get the largest gain c i · x i , Title Page i =1 – we should allocate all the resources X 0 to the task ◭◭ ◮◮ with the largest gain c i per unit resource. ◭ ◮ • In this case, the resulting gain is equal to X 0 · max i =1 ,...,m c i . Page 11 of 14 Go Back Full Screen Close Quit
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