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Note that we use NumericQ for all of the input arguments to lhs. This is important when we plot the results.

The case is covered where delta is 0 for a particular parameter.

Now I'll plot the results for the first set of parameters. For the large spread in the s values, 20.3 to 23.1 I had to increase the delta's or the min and max were too close to be visible.

and indeed testing shows that it does indeed produce a smaller minimum:

Note that we use NumericQ for all of the input arguments to lhs. This will be required the results are plotted.

The case is covered where delta is zero for a particular parameter.

Now plot the results for the first set of parameters. For the large spread in the s values the delta's had to be increased or the min and max were too close to be visible.

and indeed testing shows that it does produce a smaller minimum:

Numerical Derivatives and a faster version

If one were able to ascertain the polarity that a change in a parameter has on the outcome one could construct a faster version by reducing the number of permutations.

testDeriv = With[
  {
   a1 = -23.29,
   b1 = 0.88,
   c1 = 20.82
   },
  Table[
   {s, (lhs[s, a1 + 0.001, b1, c1] - lhs[s, a1, b1, c1])/0.001,
    (lhs[s, a1, b1 + 0.001, c1] - lhs[s, a1, b1, c1])/0.001,
    (lhs[s, a1, b1, c1 + 0.001] - lhs[s, a1, b1, c1])/0.001},
   {s, 20.3, 23.3, 0.5}
   ]
  ]

(* {{20.3, 0.00672811, 0.00102687, 0.0117371}, {20.8, 
  0.00194452, -0.000360691, 0.00494013}, {21.3, 
  0.000116524, -0.0000671435, 0.000553761}, {21.8, 
  7.0401*10^-8, -7.15508*10^-8, 8.08399*10^-7}, {22.3, 
  2.90222*10^-17, -4.32226*10^-17, 9.63237*10^-16}, {22.8, 
  4.69431*10^-46, -9.28372*10^-46, 5.11146*10^-44}, {23.3, 
  3.58277*10^-136, -8.85485*10^-136, 1.5721*10^-133}} *)

It appears that the derivative of lhs with respect to a and c is always positive and with respect to b may be either positive or negative.

Using that hypothesis a faster version of the lhs for intervals would be:

lhsMinus2[s_?NumericQ, a_?NumericQ, da_?NumericQ, b_?NumericQ, 
  db_?NumericQ, c_?NumericQ, dc_?NumericQ] := Module[
  {
   aList,
   bList,
   cList
   },
  aList = If[da == 0, {a}, {a - da}];
  bList = If[db == 0, {b}, {b, b - db, b + db}];
  cList = If[dc == 0, {c}, {c - dc}];
  
  Min[Map[lhs[s, Sequence @@ #] &, 
    Flatten[Outer[List, aList, bList, cList], 2]]]
  ]

and indeed testing shows that it does indeed produce a smaller minimum:

testMinus = With[
   {
    a1 = -23.29,
    da = 0.04,
    b1 = 0.88,
    db = 0.04,
    c1 = 20.82,
    dc = 0.01
    },
   Table[
    {s, lhsMinus[s, a1, da, b1, db, c1, dc], 
     lhsMinus2[s, a1, da, b1, db, c1, dc]},
    {s, 20.3, 23.3, 0.5}
    ]
   ];

Map[{#[[1]], #[[3]]/#[[2]]} &, testMinus]
(* {{20.3, 1.}, {20.8, 0.98248}, {21.3, 0.947516}, {21.8, 
  0.909672}, {22.3, 0.870701}, {22.8, 0.832196}, {23.3, 0.794958}} *)

The same is true for the positive side:

lhsPlus2[s_?NumericQ, a_?NumericQ, da_?NumericQ, b_?NumericQ, 
  db_?NumericQ, c_?NumericQ, dc_?NumericQ] := Module[
  {
   aList,
   bList,
   cList
   },
  aList = If[da == 0, {a}, {a + da}];
  bList = If[db == 0, {b}, {b, b - db, b + db}];
  cList = If[dc == 0, {c}, {c + dc}];
  
  Max[Map[lhs[s, Sequence @@ #] &, 
    Flatten[Outer[List, aList, bList, cList], 2]]]
  ]

testPlus = With[
  {
   a1 = -23.29,
   da = 0.04,
   b1 = 0.88,
   db = 0.04,
   c1 = 20.82,
   dc = 0.01
   },
  Table[
   {s, lhsPlus[s, a1, da, b1, db, c1, dc], 
    lhsPlus2[s, a1, da, b1, db, c1, dc]},
   {s, 20.3, 23.3, 0.5}
   ]
  ];

Map[{#[[1]], #[[3]]/#[[2]]} &, testPlus]
(* {{20.3, 1.}, {20.8, 1.01669}, {21.3, 1.05383}, {21.8, 
  1.09744}, {22.3, 1.14645}, {22.8, 1.19945}, {23.3, 1.25562}} *)

With[
 {
  a1 = -23.29,
  da = 0.08,
  b1 = 0.88,
  db = 0.08,
  c1 = 20.82,
  dc = 0.04
  },
 LogPlot[
  {
   lhsMinus[s, a1, da, b1, db, c1, dc],
   lhs[s, a1, b1, c1],
   lhsPlus[s, a1, da, b1, db, c1, dc]
   },
  {s, 20.3, 23.1},
  {PlotStyle -> {Blue, Black, Red}}
  ]
 ]

With[
 {
  a1 = -23.29,
  da = 0.08,
  b1 = 0.88,
  db = 0.08,
  c1 = 20.82,
  dc = 0.04
  },
 LogPlot[
  {
   lhsMinus[s, a1, da, b1, db, c1, dc],
   lhs[s, a1, b1, c1],
   lhsPlus[s, a1, da, b1, db, c1, dc]
   },
  {s, 20.3, 23.1},
  {PlotStyle -> {Blue, Black, Red}},
   PlotRange -> {10^-35, 1}
  ]
 ]