Questions Tagged With minimaxhttp://www.or-exchange.com/tags/minimax/?type=rssquestions tagged <span class="tag">minimax</span>enThu, 11 Aug 2016 09:14:00 -0400Linearize Minimax and Minimin in Objective function?http://www.or-exchange.com/questions/14110/linearize-minimax-and-minimin-in-objective-function<p>Dear OR-Exchange Experts,</p>
<p>I would like to know if it is possible to linearize the following objective function:</p>
<p>$$\text{minimize } Z(\textbf{x}) = m_1 * x_1 + m_2 * x_2 + c_1 * \max((x_1 + x_2) - C_{\epsilon}, 0) - c_2 * \min((x_1 + x_2) + C_{\chi}, 0)$$</p>
<p>subject to</p>
<p>$$L_1 \le x_1 \le U_1$$</p>
<p>$$L_2 \le x_2 \le U_2$$</p>
<p>\(L\) and \(U\) are the lower and the upper bound. \(x_1\), \(x_2\) can adopt any real value between the bounds (either negative or positive values)</p>
<p>where the following parameters are constant and positive, i.e. $$m_1, m_2, c_1, c_2, C_{\epsilon}, C_{\chi} \ge 0$$.</p>
<p>My first approach is to move the maximum and minimum into a constraint with two new continuous variables of a special ordered set of type 1:
$$\{k_1, k_2\}$$ with exactly that order.</p>
<p>and add the following two constraints:
$$k_1 \ge c_1 * ((x_1 + x_2) - C_{\epsilon})$$
$$k_2 \ge -c_2 * ((x_1 + x_2) + C_{\chi})$$</p>
<p>such that the objective function becomes
$$\text{minimize } Z(\textbf{x}) = m_1 * x_1 + m_2 * x_2 + k_1 + k_2$$</p>
<p>I am not sure if it is not too complex. Maybe someone of you know some hints or a better way. I am thankful for any help. Thank you.</p>ThomasThu, 11 Aug 2016 09:14:00 -0400http://www.or-exchange.com/questions/14110/linearize-minimax-and-minimin-in-objective-functionprogrammingminimaxminiminlinearMinimax two-stage stochastic program questionhttp://www.or-exchange.com/questions/13991/minimax-two-stage-stochastic-program-question<p>Dear or-exchange community,</p>
<p>I have what might seem like a trivial question to the most of you, but it puzzles me and I cannot figure it the answer out with complete certainty.</p>
<p>I am dealing with 2-stage stochastic programming and I solve programs that have focused on the expected cost so far. Now, I am planning to focus a bit on the minimax side of things.</p>
<p>Let's assume that I have a 2-stage SP problem, where in my 1st stage I have to make an investment decision for equipment and for its capacity between two options. In the second stage the operation of this equipment is set to meet some demand.</p>
<p>What is uncertain is the cost of operation for each equipment, which I assume consists of three discrete scenarios with known probabilities for each. Therefore, in total, I can create nine combinations of scenarios with their probabilities.</p>
<p>It's clear, therefore, to me how to formulate the expected cost problem. For the minimax problem, now, do I have to consider all scenarios again and minimize the worst case or would it suffice to directly take the scenario that gives me the highest cost for both pieces of equipment and solve directly for that?</p>
<p>Of course in this case I will not know how to operate it in the other scenarios, but I will know my worst-case outcome, which is what I am interested in. </p>
<p>Is this the notion behind minimax problems? Or should I still consider all scenarios, multiply each with their probability and then minimize the worst outcome with probability-weighted scenarios? </p>
<p>Does all this make any sense? Any feedback will be appreciated.</p>
<p>Regards,</p>
<p>George M.</p>gmavromWed, 06 Jul 2016 14:17:43 -0400http://www.or-exchange.com/questions/13991/minimax-two-stage-stochastic-program-questionstochasticminimaxtwo-stageMaximin Optimization Solverhttp://www.or-exchange.com/questions/13181/maximin-optimization-solver<p>Hi, all. I have a max-min program with linear objective and linear constraints for which Matlab global optimization solver is inaccurate and too slow. Can someone please recommend other decent solvers?</p>NNINGMon, 04 Jan 2016 18:17:01 -0500http://www.or-exchange.com/questions/13181/maximin-optimization-solverminimaxoptimizationmachine-learning[ANN] Introducing Minimax: an LP/MILP solver for iPhonehttp://www.or-exchange.com/questions/7029/ann-introducing-minimax-an-lpmilp-solver-for-iphone<p>Based on some ideas we discussed on this site ages ago, I decided to make an LP/MILP solver for the iPhone. It was quite fun learning iOS programming, and also figuring out what the interface for a solver should be when designing for such a small screen, and for mobile.</p>
<p>I also had some ideas for a simplified modelling language that would be easy to learn the basics of, but which could also be flexible - the result is OptML, which is used within the app. It can also import MPS models, which are then converted to OptML.</p>
<p><a href="http://simplexify.net/blog/2012/12/15/introducing-minimax-an-lp-solver-for-iphone.html">Introductory blog post</a><br>
<a href="http://simplexify.net/minimax">Minimax product page</a><br>
<a href="http://simplexify.net/optml">OptML page</a></p>DC WoodsSat, 15 Dec 2012 00:43:47 -0500http://www.or-exchange.com/questions/7029/ann-introducing-minimax-an-lpmilp-solver-for-iphonemodeling-languagesminimaxsolversiphoneoptml