# Minimize sum of absolute values linear programming

If the argument x is complex or the function fun returns complex residuals, it must be wrapped in a real function of real arguments, as shown at the end of the Examples section. Initial guess on independent variables. If float, it will be treated as a 1-D array with one element. Method of computing the Jacobian matrix an m-by-n matrix, where element i, j is the partial derivative of f[i] with respect to x[j].

The keywords select a finite difference scheme for numerical estimation. Lower and upper bounds on independent variables. Defaults to no bounds. Each array must match the size of x0 or be a scalar, in the latter case a bound will be the same for all variables. Use np. Generally robust method. Not recommended for problems with rank-deficient Jacobian. Usually the most efficient method for small unconstrained problems.

Tolerance for termination by the change of the cost function. Default is 1e Tolerance for termination by the change of the independent variables. The exact condition depends on the method used:. Tolerance for termination by the norm of the gradient. The exact condition depends on a method used:. Characteristic scale of each variable. Gives a standard least-squares problem.It can be observed that it is best to have b as the median of the modified array B[].

So, after sorting array B[], the problem can be solved following the steps given below. Below is the implementation of the above approach. Skip to content. Change Language. Related Articles. Table of Contents. Improve Article.

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Save Article. Like Article. Last Updated : 20 Jul, Java program for above approach. Modify the array. Stores the tatti shayari in punjabi answer. Update answer. Return the answer. This code is contributed by Parth Manchanda. Sort arr. Abs arr[i] - b. Write ans. Previous Make all array elements equal by repeatedly replacing largest array element with the second smallest element.

Recommended Articles. Minimize sum of absolute differences between given Arrays by replacing at most 1 element from first Array.

Minimize sum of absolute differences of same-indexed elements of two given arrays by at most one replacement. Minimum number of swaps required to minimize sum of absolute differences between adjacent array elements. Minimize sum of absolute difference between all pairs of array elements by decrementing and incrementing pairs by 1. Minimize cost to empty given array where cost of removing an element is its absolute difference with Time instant. Generate an N-length array with sum equal to twice the sum of its absolute difference with same-indexed elements of given array.

Minimize absolute difference between the smallest and largest array elements by minimum increment decrement operations. Minimize remaining array element by removing pairs and replacing them by their absolute difference. Minimize array sum by replacing greater and smaller elements of pairs by half and double of their values respectively atmost K times.

Find unique lexicographically increasing quadruplets with sum as B and GCD of absolute values of all elements is 1. Decimal equivalent of concatenation of absolute difference of floor and rounded-off values of array elements as a Binary String. Absolute difference between floor of Array sum divided by X and floor sum of every Array element when divided by X.

Sum of absolute differences of pairs from the given array that satisfy the given condition. Minimize the difference between the maximum and minimum values of the modified array.

Minimize count of array elements to be removed such that at least K elements are equal to their index values. Minimum absolute value of K — arr[i] for all possible values of K over the range [0, N — 1].So what if absolute values must be formulated:.

This is the easiest case. If abs X must be equal to zero, then this can only be fulfilled if X is zero. So the condition can also be written as:. This also implies that maximum is always bigger than or equal to zero. Else the constraint would always be impossible mathematical impossible with real numbers.

And the fortunate thing is that if we need the one restriction that the other is always redundant. If X is positive, then -X is negative and thus always less than maximum which is always positive, remember and thus the second equation is then redundant. If X is negative, then it is always less than maximum which is always positive, remember and thus the first equation is then redundant. This can also be seen easily from the graphical representation.

So just add the following two equations:. This also implies that minimum should always be bigger than zero. Else the constraint would always be fulfilled and there is no point in having the constraint. Unfortunately, the trick as for a maximum cannot be used here. If X is positive, then -X is not greater than minimum, in contrary It can also be seen from the graphical representation that this restriction is discontinue.

This has as effect that it is not possible to convert this into a set of linear equations. A possible approach to overcome this is making use of integer variables. In particular by using a binary variable B:. M is a large enough constant. See later. The binary variable B takes care of the discontinuity.

It can be either 0 or 1. With M large enough, this makes one or the other constraint obsolete. X must be positive and larger than minimum. With M large enough, the second constraint is always fulfilled. X must be negative and -X be larger than minimum. With M large enough, the first constraint is always fulfilled. It is important to use a realistic value for M. Don't use for example 1e30 for it.

This creates numerical instabilities and even if not then tolerances will give problem. Because of tolerances, B may not be zero, but actually for example 1e This multiplied with 1e30 gives not zero, but 1e10!

Not what was mathematically formulated! So how big must M be? Well, we can make a prediction. If we can predict how large X can become absolutelythen we can predict a maximum value needed for M for this to work.MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up. Is is possible to model this as a standard linear program without integer variables and extensions like disjunctions.

This blog suggests several solutions. The solution with SOS2 method appears to work sometimes in glpk and coin-or, though sometimes it doesn't work. In general such a problem is NP-hard, so not expressible by a polynomially-sized linear program. Sign up to join this community.

The best answers are voted up and rise to the top. Maximizing linear objective function with absolute values Ask Question. Asked 7 years, 10 months ago. Active 7 years ago. Viewed 3k times. I have a linear program over the reals and don't want to introduce integer or binary variables. What I know. Minimizing sum of absolute values is possible. The lpsolve documentation suggests using a binary variable. If this is impossible, there might be reduction from a NP-hard problem.

Improve this question. If you have a typical polyhedral feasible region, then maximizing the sum of absolute values is in general nonconvex, hence not a linear program which are convex.

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Related 0. Question feed. MathOverflow works best with JavaScript enabled.The method of least absolute deviation LAD finds applications in many areas, due to its robustness compared to the least squares regression LSR method. LAD is robust in that it is resistant to outliers in the data. This may be helpful in studies where outliers may be ignored. Since LAD is nonsmooth optimization problem, this paper proposed a metaheuristics algorithm named novel global harmony search NGHS for solving.

Least squares regression LSR method is one of the oldest and the most widely used statistical tools for linear models. Its theoretical properties have been extensively studied and are fully understood. Despite its many superior properties, the LSR estimate can be sensitive to outliers and, therefore, nonrobust [ 1 ].

In order to overcome these problems, the researchers have recently investigated the alternative regression method which is least absolute deviation LAD method. Least absolute deviation LADalso known as least absolute errors LAEleast absolute value LAVor least absolute residual LARor the norm problem, is a mathematical optimization technique similar to the LSR that attempts to find a function which closely approximates a set of data.

This may be helpful in studies where outliers may be safely and effectively ignored. Despite its long history and many ground-breaking works, the LAD has not been explored in theory as well as in application to the extent as the LSR [ 3 ]. This is largely because computing LAD estimates are more difficult than computing LSR estimates due to the fact that algorithmic method must be employed to calculate LAD estimates.

Over the past few years, a number of approaches have been developed for solving the LAD using classical mathematical programming methods. Since Charnes et al. Due to these developments in theoretical and computational aspects, the LAD method has become increasingly popular. In particular, it has many applications in econometrics and biomedical studies; see Bassett and Koenker [ 9 ], Powell [ 10 ], Buchinsky [ 11 ], among many others.

The NGHS algorithm includes two important operations: position updating and genetic mutation with a small probability. The former enables the worst harmony of harmony memory to move to the global best harmony rapidly in each iteration, and the latter can effectively maintain population diversity and prevent the NGHS from trapping into the local optimum. The remaining sections of this chapter are organized as follows.

In Section 2LAD model is shown. Numerical results are presented and compared in Section 5.In this chapter we discuss various aspects of linear optimization. We first introduce the basic concepts of linear optimization and discuss the underlying geometric interpretations.

We then give examples of the most frequently used reformulations or modeling tricks used in linear optimization, and finally we discuss duality and infeasibility theory in some detail. The most basic type of optimization is linear optimization. In linear optimization we minimize a linear function given a set of linear constraints. For example, we may wish to minimize a linear function. The function we minimize is often called the objective function ; in this case we have a linear objective function.

The constraints are also linear and consist of both linear equalities and inequalities. We typically use more compact notation. The domain where all constraints are satisfied is called the feasible set ; the feasible set for 2. Linear optimization problems are typically formulated using matrix notation. The standard form of a linear minimization problem is:. For example, we can pose 2. There are many other formulations for linear optimization problems; we can have different types of constraints.

All these formulations are equivalent in the sense that by simple linear transformations and introduction of auxiliary variables they represent the same set of problems.

The polyhedral description of the feasible set gives us a very intuitive interpretation of linear optimization, which is illustrated in Fig. The polyhedron shown in the figure is nonempty and bounded, but this is not always the case for polyhedra arising from linear inequalities in optimization problems. In such cases the optimization problem may be infeasible or unbounded, which we will discuss in detail in Sec.

In this section we present useful reformulation techniques and standard tricks which allow constructing more complicated models using linear optimization. It is also a guide through the types of constraints which can be expressed using linear in equalities. The function is defined as the maximum of 3 affine functions.

Piecewise-linear functions have many uses linear in optimization; either we have a convex piecewise-linear formulation from the onset, or we may approximate a more complicated nonlinear problem using piecewise-linear approximations, although with modern nonlinear optimization software it is becoming both easier and more efficient to directly formulate and solve nonlinear problems without piecewise-linear approximations.

Clearly 2. Therefore, we can model 2. For example, suppose we are given an underdetermined linear system. The basis pursuit problem. Using 2. Using Sec. Perhaps surprisingly, it can be turned into a linear problem if we homogenize the linear constraint, i.For more information, contact your sales or technical support representative.

For keyword LAV defaultthe criterion satisfied is the minimization of the sum of the absolute values of the deviations of the observed response y i from the fitted response:. Under this criterion, known as the L 1 or LAV least absolute value criterion, the regression coefficient estimates minimize:. The estimation problem can be posed as a linear programming problem. The special nature of the problem, however, allows for considerable gains in efficiency by the modification of the usual simplex algorithm for linear programming.

These modifications are described in detail by Barrodale and Roberts In many cases, the algorithm can be made faster by computing a least-squares solution prior to the use of keyword Lav. This is particularly useful when a least- squares solution has already been computed. The procedure is as follows:. When multiple solutions exist for a given problem, option LAV may yield different estimates of the regression coefficients on different computers, however, the sum of the absolute values of the residuals should be the same within rounding differences.

The informational error indicating nonunique solutions may result from rounding accumulation. Conversely, because of rounding the error may fail to result even when the problem does have multiple solutions.

Frequencies are useful if there are repetitions of some observations in the data set. In general, keyword Llp minimizes the L p norm:. In the discussion that follows, we will first present the algorithm with frequencies and weights all taken to be one. Later, we will present the modifications to handle frequencies and weights different from one. The cutoff value of 1. Otherwise, the residuals from the k -th iteration:. Specifically, we modify the definition of:. If the first attempted step does not lead to a decrease of at least one-tenth of the predicted decrease in the p -th power of the L p norm of the residuals, a backtracking linesearch procedure is used.

The backtracking procedure uses a one-dimensional quadratic model to estimate the backtrack constant p. The value of p is constrained to be no less that 0.

An approximate upper bound for p is 0. Convergence is declared when the maximum relative change in the residuals from one iteration to the next is less than or equal to EPS. The relative change:. I need a linear program to minimize the sum of several absolute values, but the inclusion of an absolute value means the linear solver won't work.

bedenica.eu › bedenica.eu › Optimization_wit. Optimization with absolute values is a special case of linear programming in which a problem made. minimization of the sum of absolute values; minimization of the largest absolute value; and maximization of a fraction. Approximations include grid point. Or, we could minimize the sum of the absolute values of the errors (least absolute deviations).

I used the latter—it provides answers that are. Minimizing sum of absolute values is possible. such a problem is NP-hard, so not expressible by a polynomially-sized linear program. As an alternative to Warren's solution involving 2^n constraints for a sum of n absolute value terms, one could introduce n extra variables.

bedenica.eu › title=Optimization_with_absolute_values. Absolute values can exist in linear optimization problems in two Positive for a minimization problem; Negative for a maximization. Minimizing the Sum of Absolute Deviations. In some applications, we are given several linear objective functions of the decision variables.

I am trying to solve a linear program with command "linprog". The inequality constrains has a sum of varaibles. e.g. I have 3 variable. x1,x2,x3. Hello all, I am trying to solve a problem based on some computer programming task I am trying to solve, and I have encountered a situation I.

Is there a way to maximize sum of two absolute values? there's no way to convert maximization of absolute values into a linear optimization problem that. Minimize sum of absolute values of A[i] – (B + i) for a given array · Traverse the array arr[ ] and decrease every element by their index (i + 1). a mixed integer programming formulation.

Heuristic solution methods are developed and the performance of each heuristic is compared with the optimal. The recent interest for this problem can be explained since frequently occurring optimization problems such as linear complementarity problem or.

obj. z=e=sum(j, abs(x(j))); cons(i).

## Novel Global Harmony Search Algorithm for Least Absolute Deviation

sum(j, a(i,j)*x(j)) =l= b(i); model foo /all/; solve foo minimizing z using lp. Chapter 6. Linear Programming Tricks. Instead of the standard cost function, a weighted sum of the absolute values of the variables is to be minimized.

The method minimizes the sum of absolute errors (SAE) (the sum of the absolute values of the vertical "residuals" between points. Minimize the Weighted Sum of Absolute Values Optimization of the sum of weighted absolute values can only be performed under certain conditions with.