• Georgia Institute Of Technology

• ISYE 6644

OPTIMIIZATION Cons: no software to build b/c req. human insight expertise, takes lots of prac. to learn Opt. 4 presscriptive analy.(airplane mechanic scheduling, GPS routing for cars, army recruit asset usage, diet prob.) OPT. Elements: dec.var.: dec. the opt. solver will pick the best values for |constraints: restrictions on those dec. we made(need 2 has a relation be refl. by constraint) | obj. func.: meas. quality of sol., a set of values for dec. var. which we trying to max. or min. (can have alpha, beta, factor d)| feasible sol: set of values 4 dec.var. that satify all constrain in opt. prob.| opt. sol: feas. sol. that has best obj.func.values) How 2 create dec.var.: things we can make about/change, cant create dec.var.on a fact) How 2 create constraints: 1.create const. to limit poss. sol. | 2. the opt. prob., using ineq. |3. equality to des. Interactions bet. Dec.var. |4.cant create fact const.| 5. x_I>-0(non-negativity) Diet Prob: Call Center Prob: Stock Market Prob: OPT. For STAT. MODELS: notion def. for stat. Models(x_ij: var. that changes, a_1:constant coeff. vice versa) Lin. Reg. : Obj.Funt:(min. sum of n data pts i Lasso. Reg. : stand.reg w/ 1 constraint add to restrict sum of var. ridge Reg. : restriction on sum of sq. var. Elastic : combine 2 terms into 1 const. Log. Reg. : var. r reg. coeff, OF(min. pred.err) Hard class. : const(each data pt need to correct class, OF;(min. margin bet. SV) soft class. : OF(min.margin & class. Err.) Expo. smoothing: OF(min.err.) ARIMA & GARCH OF(min.pred.err.) CLASS. OPT. MODELS: Lin. Prog.:(uses in delivery serv, fin. Inst. Convex Prog.:(uses: SVM, lin.reg., engin. prob) Convex opt., integer, & gen. non-convex prog. Lin.prog, convex prog, network-> guranteed opt. sol. while integer & non-convex end up w/ infeasible sol.(no sol. satisfy constraint)/ local optimum(sol. Is != best) network prog.: a type of lin. prog(solve quicker than lin. prog., usage: GPS routing shortest path, assign. prob.(worker vs job), max. flow( the amt of how much oil can get via pipe.) STOCHASTIC OPT.: We implicitly assume we knew all input of data, if there is uncertainty & random we use 1. model conservatively method 2. Scenario modeling:create diff. scenarios&optimize all of them(use heuristic) Def robust sol: worse-case outcome over all poss. Scenarios is least bad Dynamic program: diff.struc/use states(specify what’s going on in the sys., e.g. status of a bug detect & fixed), then @ each state, dec.-maker makes a decision, then sys. move to that state / can use bellman’s eq. to deter. opt.dec. to make @ every state, gives no uncertainty Stochastic Dynamic program:can change dyn.prog. model by incorporate w/ prob. If Downloaded by Jesus Rincon (chechu717@yahoo.com) lOMoARcPSD|53162341 we know the prob. of going from 1 state to another Markove dec. Process: if we have discret(fixed or countable) set of states & dec, prob. only depend on curr. state & decision HOW OPT. ALGO. WORKS Two main steps(1. initialization:start w/ a sol. that can be simple, bad, feasi.) / 2. repeat(start w/ curr. sol. ->find improving direction(t) --> add step size to theta --> new sol.=old sol. + improving direction(t)*step size(theta, which will be change) -> keep repeat ->sol. doesnt change/run out of time Newton’s eq.:( above steps similar to this eq. on finding root of f(x), @ step n, a curr. sol., find improving direction & our new sol. : xn+1=xn-1*t(might gurantee to find opt.sol) Global max.:OB(max) vs global min.(OB->min) PROB-BASED MODELS Prob. Dist.: desc. all poss. values&likelihood of a ran.var. can take within a given range(1. f(x) must be non-neg. 4 each value of ran.var. & 2. prob. of particular range must be bet.0 to 1) Bernoulli: 1. use to model a single event or bin. Dist. W/ single trial & can only has 2 poss. Outc.(success/failure)&evt indepent | 2.e.g. single coin toss Binomial dist.: consist of single evt repeated mult. times, 2 poss. outcomes, #s of trials fixed, evt indepdent, prob. of success is same in each trial / diff. from bernoulis: if each bernoulli trial is independent, #s of succ. in bern. has a bin., if bern. dist. Is bin. dist. w/ n=1 |2. #s of succ. sales calls) -->due to central limit theorem, when n inc., n trials & prob. P succ. will get closer to norm.dist., e.g. think of as survey the whole pop. -->convert into norm. Geo. Dist.:1. #s of failures be4 u get a succ. in a series of bern. trials&has 2 poss. outc., trials independent & prob. of succ. is