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468 PATH 4.6 sets i canning plants, j markets ; parameter s(i) capacity of plant i in cases, d(j) demand at market j in cases, c(i,j) transport cost in thousands of dollars per case ; $include transmcp.dat positive variables x(i,j) shipment quantities in cases p_demand(j) price at market j p_supply(i) price at plant i; equations supply(i) demand(j) rational(i,j); observe supply limit at plant i satisfy demand at market j supply(i) .. s(i) =g= sum(j, x(i,j)) ; demand(j) .. sum(i, x(i,j)) =g= d(j) ; rational(i,j) .. p_supply(i) + c(i,j) =g= p_demand(j) ; model transport / rational.x, demand.p_demand, supply.p_supply /; solve transport using mcp; Figure 28.1: A simple MCP model in GAMS, transmcp.gms
PATH 4.6 469 or alternatively positive variable z; Further information on the GAMS syntax can be found in [35]. Note that GAMS requires the modeler to write F(z) =g= 0 whenever the complementary variable is lower bounded, and does not allow the alternative form 0 =l= F(z). 1.1.2 Extension: Model Generalization While many interior point methods for linear programming exploit this complementarity framework (so-called primal-dual methods [37]), the real power of this modeling format is the new problem instances it enables a modeler to create. We now show some examples of how to extend the simple model (28.2) to investigate other issues and facets of the problem at hand. Demand in the model of Figure 28.1 is independent of the prices p. Since the prices p are variables in the complementarity problem (28.2), we can easily replace the constant demand d with a function d(p) in the complementarity setting. Clearly, any algebraic function of p that can be expressed in GAMS can now be added to the model given in Figure 28.1. For example, a linear demand function could be expressed using ∑ x i,j ≥ d j (1 − p d j ), ∀j. i:(i,j)∈A Note that the demand is rather strange if p d j exceeds 1. Other more reasonable examples for d(p) are easily derived from Cobb-Douglas or CES utilities. For those examples, the resulting complementarity problem becomes nonlinear in the variables p. Details of complementarity for more general transportation models can be found in [13, 16]. Another feature that can be added to this model are tariffs or taxes. In the case where a tax is applied at the supply point, the third general inequality in (28.2) is replaced by p s i (1 + t i ) + c i,j ≥ p d j , ∀(i, j) ∈ A. The taxes can be made endogenous to the model, details are found in [35]. The key point is that with either of the above modifications, the complementarity problem is not just the optimality conditions of a linear program. In many cases, there is no optimization problem corresponding to the complementarity conditions. 1.1.3 Nonlinear Complementarity Problem We now abstract from the particular example to describe more carefully the complementarity problem in its mathematical form. All the above examples can be cast as nonlinear complementarity problems (NCPs) defined as follows: (NCP) Given a function F : R n → R n , find z ∈ R n such that 0 ≤ z ⊥ F (z) ≥ 0. Recall that the ⊥ sign signifies that one of the inequalities is satisfied as an equality, so that componentwise, z i F i (z) = 0. We frequently refer to this property as z i is “complementary” to F i . A special case of the NCP that has received much attention is when F is a linear function, the linear complementarity problem [8]. 1.2 Walrasian Equilibrium A Walrasian equilibrium can also be formulated as a complementarity problem (see [33]). In this case, we want to find a price p ∈ R m and an activity level y ∈ R n such that 0 ≤ y ⊥ L(p) := −A T p ≥ 0 0 ≤ p ⊥ S(p, y) := b + Ay − d(p) ≥ 0 (28.3)
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GAMS — The Solver Manuals c○ 20
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4 CONTENTS
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6 Basic Solver Usage Option iterlim
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8 Basic Solver Usage keyword(s) [mo
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10 AlphaECP 1.2 Running GAMS/AlphaE
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12 AlphaECP A: Infeasible, deleted
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14 AlphaECP 0 Never. 1 Call the NLP
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16 AlphaECP mipoptcr (real) Relativ
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18 AlphaECP Westerlund T. and Pette
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20 BARON 1.1 Licensing and software
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22 BARON on probing : 000:00:00, in
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24 BARON 4.2 Finding the best, seco
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26 BARON in the form of solver boun
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28 BARON 5.4 Range reduction option
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30 BARON 5.6 Heuristic local search
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32 BARON Option Description Default
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34 BDMLP
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36 BENCH option modeltype=bench; Th
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38 BENCH Option Description Default
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40 BENCH In the example below, EXAM
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42 BENCH --- Spawning solver : CPLE
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44 BENCH To terminate not only the
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46 BENCH
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48 COIN-OR also found their way int
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50 COIN-OR This sets the algorithm
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52 COIN-OR Option type default B-BB
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54 COIN-OR integer tolerance: Set i
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56 COIN-OR stop diving on cutoff: F
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58 COIN-OR userjobid: Postfixes gdx
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60 COIN-OR nlp solve max depth: Set
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62 COIN-OR maxmin crit no sol: Weig
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64 COIN-OR General Options iterlim
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66 COIN-OR This option causes GAMS
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68 COIN-OR scaling (string) Scaling
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70 COIN-OR sos This option let CBC
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72 COIN-OR knapsackcuts (string) De
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74 COIN-OR 1 Turns the feasibility
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76 COIN-OR userheurmult (integer) D
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78 COIN-OR This option determines t
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80 COIN-OR noiterlim (integer) Allo
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82 COIN-OR linear_solver pardiso pa
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84 COIN-OR # This is a comment # Tu
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86 COIN-OR acceptable tol: ”Accep
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88 COIN-OR slack bound push: Desire
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90 COIN-OR linear system scaling: M
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92 COIN-OR mumps pivtol: Pivot tole
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94 COIN-OR fixed variable treatment
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96 COIN-OR adaptive mu monotone ini
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98 COIN-OR quality function balanci
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100 COIN-OR alpha min frac: Safety
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102 COIN-OR • no: don’t skip
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104 COIN-OR max resto iter: Maximum
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106 COIN-OR • no: perturbation on
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108 COIN-OR 6.2 Usage of CoinScip T
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110 COIN-OR gams/usergdxname (strin
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112 CONOPT 1 Introduction Nonlinear
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114 CONOPT 130 0 103 2.1776589484E+
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116 CONOPT The two messages above t
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118 CONOPT CONOPT time Total 0.109
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120 CONOPT 6 Hints on Good Model Fo
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122 CONOPT by the reduced complexit
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124 CONOPT of 4.1**2, which means t
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126 CONOPT X.SCALE(I,J,K)$IJK(I,J,K
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128 CONOPT delta. The error is redu
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130 CONOPT the NLP solver will comp
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132 CONOPT The relationship between
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134 CONOPT A : ❅ ❅ ❅ ❅ Zero
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136 CONOPT EQUATION E1, E2, E3, E4;
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138 CONOPT unfortunate that a redun
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140 CONOPT The Crash procedure is n
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142 CONOPT infeasible equations (th
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144 CONOPT The steepest edge proced
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146 CONOPT CONOPT will after the pr
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148 CONOPT X2 appearing in E2: Pivo
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150 CONOPT Until a final solution h
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152 CONOPT Option Description Defau
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154 CONOPT Option Description Defau
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156 CONOPT
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158 CONVERT • LindoMPI • LINGO
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160 CONVERT Option Description Defa
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162 CPLEX 11 2 How to Run a Model w
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164 CPLEX 11 • You can collect al
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166 CPLEX 11 individual GDX solutio
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168 CPLEX 11 parallelmode predual p
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170 CPLEX 11 mipordind mipordtype m
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172 CPLEX 11 6 Special Notes 6.1 Ph
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174 CPLEX 11 Start. The number of t
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176 CPLEX 11 Aggregator did 30 subs
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178 CPLEX 11 Nodes Cuts/ Node Left
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180 CPLEX 11 baritlim (integer) Det
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182 CPLEX 11 -1 Do not generate cov
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184 CPLEX 11 .divflt (real) A diver
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186 CPLEX 11 feasoptmode (integer)
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188 CPLEX 11 implbd (integer) Deter
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190 CPLEX 11 wasted computation tim
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192 CPLEX 11 netppriind (integer) N
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194 CPLEX 11 Settings of this paral
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196 CPLEX 11 1 Force node presolve
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198 CPLEX 11 repairtries (integer)
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200 CPLEX 11 simdisplay (integer) T
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202 CPLEX 11 populate procedure aga
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204 CPLEX 11 1 Display standard min
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206 CPLEX 11
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208 DECIS 1 DECIS 1.1 Introduction
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210 DECIS 1.4 Solving the Universe
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212 DECIS SMPS (stochastic mathemat
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214 DECIS statement would work as w
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216 DECIS RHS demand m 1000.00 peri
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218 DECIS omega1 defines all realiz
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220 DECIS nzrows — Number of rows
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222 DECIS Example The following exa
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224 DECIS 20 0.2464E+05 0.2464E+05
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226 DECIS pro 0.1 0.2 0.5 0.1 0.1 ;
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228 DECIS parameter hm2(dl) / h 100
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230 DECIS 15. ERROR: error code: in
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232 DECIS REFERENCES DECIS License
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234 DICOPT Although the algorithm h
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236 DICOPT where λ i is the Lagran
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238 DICOPT option rminlp=minos; sol
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240 DICOPT **** ERRORS(S) IN EQUATI
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242 DICOPT * stop only on infeasibl
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244 DICOPT mipoptfile s 1 s 2 . . .
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246 DICOPT nlptracelevel 3 As nlptr
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248 DICOPT --- DICOPT: Checking con
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250 DICOPT NLP 2 1.72097< 0.00 3 0
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252 DICOPT • The GAMS option stat
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254 DICOPT REFERENCES
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256 EMP
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258 EXAMINER The optimality checks
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260 EXAMINER Option Description Def
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262 EXAMINER
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264 GAMS/AMPL --- No AmplPath optio
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266 GAMS/LINGO --- No LingoPath opt
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268 KNITRO • Derivative-free, 1st
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270 KNITRO Option Description Defau
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272 KNITRO Option Description Defau
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274 KNITRO If outlev=2, information
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276 KNITRO (Dense) Quasi-Newton BFG
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KNITRO References [1] R. H. Byrd, J
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280 LGO GAMS/LGO does not rely on a
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282 LGO 3 The GAMS/LGO Log File The
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284 LGO Illustrative References R.
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286 LINDOGlobal report the best one
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288 LINDOGlobal Infeasibility of be
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290 LINDOGlobal MIP AOPTTIMLIM time
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292 LINDOGlobal 4.5 NLP Options NLP
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294 LINDOGlobal 3 Decomposed model
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296 LINDOGlobal SOLVER USECUTOFFVAL
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298 LINDOGlobal 0 Solver decides 1
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300 LINDOGlobal MIP INTTOL (real) A
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302 LINDOGlobal MIP HEULEVEL (integ
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304 LINDOGlobal MIP BRANCH PRIO (in
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306 LINDOGlobal 1 Base MIP calculat
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308 LINDOGlobal POSTLEVEL (integer)
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310 LINDOGlobal
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312 LogMIP
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314 MILES When l = −∞ and u =
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316 MILES Given z, MILES uses the v
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318 MILES Pivoting Rules When z 0 e
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320 MILES Pivot Selection Option De
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322 MILES Table 2 Sample Iteration
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324 MILES INFEAS. PIVOTS IN/OUT is
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326 MILES Table 4 Status File with
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328 MILES Table 6 Status File with
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330 MILES N.H. Josephy, ”Newton
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332 MILES Table 8 Transport Model i
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334 MINOS 1 Introduction This docum
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336 MINOS their bounds: l j − δ
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338 MINOS 4 Modeling Issues Formula
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340 MINOS obj.. z =e= sum(i, sqr(re
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342 MINOS reform This option will i
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344 MINOS 6.5 Examples of GAMS/MINO
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346 MINOS B. A. Murtagh, University
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348 MINOS MINOS-Link May 25, 2002 W
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350 MINOS crash option 2 Only the c
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352 MINOS The storage required if o
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354 MINOS linear variables y or the
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356 MINOS Pivot Tolerance r Broadly
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358 MINOS Solution yes Solution no
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360 MINOS Verify option 3 A detaile
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362 MINOS The simple way to solve t
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364 MINOS REFERENCES
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366 MOSEK Furthermore, MOSEK can so
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368 MOSEK 1.5 Nonlinear Programs MO
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370 MOSEK The original problem is:
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372 MOSEK The first option specifie
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374 MOSEK MSK IPAR LOG BI output co
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376 MOSEK MSK IPAR SIM PRIMAL CRASH
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378 MOSEK MSK IPAR PRESOLVE LINDEP
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380 MOSEK MSK DPAR DATA TOL QIJ (re
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382 MOSEK MSK DPAR INTPNT CO TOL NE
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384 MOSEK MSK IPAR INTPNT STARTING
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386 MOSEK MSK IPAR SIM PRIMAL CRASH
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388 MOSEK Coefficient reduction +51
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390 MOSEK 8 The optimizer for nonco
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392 MOSEK Simplex - cpu time: 0.00
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394 MOSEK Presolve - time : 0.00 Pr
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396 MPSWRITE need to use applicatio
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398 MPSWRITE C0000004 X SAN-DIEGO N
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400 MPSWRITE 6 Using Analyze with M
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402 MPSWRITE 7 The GAMS/MPSWRITE Op
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404 MPSWRITE 8 Name Generation Exam
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406 MPSWRITE C CHICAGO T TOPEKA I c
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408 MPSWRITE the macros @i and @j.
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410 MPSWRITE SSA 3 -0.10000000E+21
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412 MPSWRITE < translate .lp file i
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414 MPSWRITE ANALYZE Version 10.0 b
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416 NLPEC gams nash MPEC=nlpec MCP=
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Page 426 and 427: 426 NLPEC L/U B equref reftype sign
Page 428 and 429: 428 NLPEC
Page 430 and 431: 430 OQNLP and MSNLP the set gathere
Page 432 and 433: 432 OQNLP and MSNLP Itn Penval Meri
Page 434 and 435: 434 OQNLP and MSNLP this guide. You
Page 436 and 437: 436 OQNLP and MSNLP Option Descript
Page 438 and 439: 438 OQNLP and MSNLP ENDDO xt ∗ =
Page 440 and 441: 440 OQNLP and MSNLP Return xt This
Page 442 and 443: 442 OSL 3 Overview of OSL OSL offer
Page 444 and 445: 444 OSL sets the amount of memory u
Page 446 and 447: 446 OSL 5.5 Examples of GAMS/OSL Op
Page 448 and 449: 448 OSL Option Description Default
Page 450 and 451: 450 OSL Option Description Default
Page 452 and 453: 452 OSL 7 Special Notes This sectio
Page 454 and 455: 454 OSL 9 Examples of MPS Files Wri
Page 456 and 457: 456 OSL XU C0000004 R0000004 0.0000
Page 458 and 459: 458 OSL Stochastic Extensions The s
Page 460 and 461: 460 OSL Stochastic Extensions 4.2 M
Page 462 and 463: 462 OSL Stochastic Extensions Optio
Page 464 and 465: 464 OSL Stochastic Extensions
Page 466 and 467: 466 PATH 4.6 3.3 Difficult Models .
Page 470 and 471: 470 PATH 4.6 $include walras.dat po
Page 472 and 473: 472 PATH 4.6 An advantage of the ex
Page 474 and 475: 474 PATH 4.6 S O L V E S U M M A R
Page 476 and 477: 476 PATH 4.6 which has a unique sol
Page 478 and 479: 478 PATH 4.6 --- Starting compilati
Page 480 and 481: 480 PATH 4.6 Code Meaning C A cycle
Page 482 and 483: 482 PATH 4.6 At the end of the log
Page 484 and 485: 484 PATH 4.6 Option Default Explana
Page 486 and 487: 486 PATH 4.6 2.6 Preprocessing The
Page 488 and 489: 488 PATH 4.6 In the context of nonl
Page 490 and 491: 490 PATH 4.6 the generated path ema
Page 492 and 493: 492 PATH 4.6 Figure 28.10: Merit Fu
Page 494 and 495: 494 PATH 4.6 INITIAL POINT STATISTI
Page 496 and 497: 496 PATH 4.6 A.1 Classical Model Th
Page 498 and 499: 498 PATH 4.6 7 orders of magnitude
Page 500 and 501: PATH References [1] S. C. Billups.
Page 502 and 503: 502 PATH REFERENCES
Page 504 and 505: 504 PATHNLP The standard GAMS model
Page 506 and 507: 506 SBB • March 21, 2001: Level 0
Page 508 and 509: 508 SBB Option Description Default
Page 510 and 511: 510 SBB Solution satisfies optcr St
Page 512 and 513: 512 SBB Non convex model! # jumps i
Page 514 and 515: 514 SCENRED running time) of the me
Page 516 and 517: 516 SCENRED eter stored in the SCEN
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518 SCENRED 6 The SCENRED Output Fi
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520 SCENRED 9 SCENRED Warnings SCEN
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522 SNOPT to be stated in the form
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524 SNOPT 2.2 Constraints and slack
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526 SNOPT By analogy with the eleme
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528 SNOPT * fill with random data x
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530 SNOPT 4 Options In many cases N
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532 SNOPT of (nonlinear) nonzeroes,
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534 SNOPT In all cases, a derivativ
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536 SNOPT • t must be a real valu
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538 SNOPT s a single line that give
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540 SNOPT • It is common for two
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542 SNOPT Unbounded objective value
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544 SNOPT --- Starting execution --
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546 SNOPT Merit is the value of the
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548 SNOPT EXIT -- Requested accurac
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550 SNOPT EXIT -- Function evaluati
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552 SNOPT The possible EXIT message
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554 PATH REFERENCES
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556 XA during the course of program
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558 XA Each XA B&B strategy has man
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560 XA Option Description Default D
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562 XA Option Description Default S
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564 XA
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566 XPRESS • option iterlim=n; or
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568 XPRESS Option Description Defau
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574 XPRESS
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